{"id":263,"date":"2017-08-23T04:34:28","date_gmt":"2017-08-23T04:34:28","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/elementary-education\/?page_id=263"},"modified":"2017-09-25T21:14:40","modified_gmt":"2017-09-25T21:14:40","slug":"measuring-triangles","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/elementary-education\/measuring-triangles\/","title":{"rendered":"Measuring Triangles"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/representational-systems\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/measuring-quadrilaterals\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Measuring Triangles<\/h1>\n<h4>Objective<\/h4>\n<p>In this next section, we\u2019ll examine some components of a triangle, and review the methods to determine the perimeter and area of triangles.<\/p>\n<p>We\u2019ll also refresh your memory about the Pythagorean Theorem (and Pythagorean triples) and delve into some basic trigonometry.<\/p>\n<h4>Previously Covered<\/h4>\n<ul>\n<li>Protractors are used to accurately measure and construct angles.<\/li>\n<li>The length of a horizontal line segment equals the difference between the x-coordinates.<\/li>\n<li>The length of a vertical line segment equals the difference between the y-coordinates.<\/li>\n<\/ul>\n<section>\n<h3>Triangles: Perimeter<\/h3>\n<p><strong>Perimeter<\/strong> is a two-dimensional measure of the distance around the figure. For any polygon, the perimeter is simply the sum of the lengths of all of its sides.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3triangle-p.jpg\" alt=\"Perimeter of a Triangle\" \/><\/center><\/p>\n<p class=\"figcaption\">Perimeter of a Triangle<\/p>\n<p>The perimeter of this triangle is 5 cm + 6 cm + 7 cm, or 18 cm.<\/p>\n<p>It\u2019s just that easy!<\/p>\n<p>Perimeter is a two-dimensional measure, so it uses units like centimeters, meters, inches, or feet.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>What is the perimeter of triangle ABC below?<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3triangle-p-question.jpg\" alt=\"Angle JKL on protractor\" \/><\/center><\/p>\n<ol>\n<li>14.5 in.<\/li>\n<li>15 in<\/li>\n<li>18 in<\/li>\n<li>Cannot be determined from the given information<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice B is correct. Since the triangle is isosceles, it has two legs that measure 4 inches each, and a base that measures 7 inches. Therefore, the perimeter is 4 in. + 4 in. + 7 in., or 15 in. Choice A is incorrect, because the segment labeled 3.5 in. is not a side of triangle ABC.<\/p>\n<\/section>\n<h3>Triangles: Area<\/h3>\n<p>The area of a two-dimensional figure is the number of square units it contains. You\u2019ve probably heard of an apartment or house being measured in square feet (ft<sup>2<\/sup>). Other examples of square units are square inches (in<sup>2<\/sup>) and square centimeters (cm<sup>2<\/sup>).<\/p>\n<p>The area of a triangle is given by the formula <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/p3_1.gif\" alt=\"A = 1\/2 b h\" \/>, where b is the base and h is the height.<\/p>\n<p>It is important to remember that the base and the height must be perpendicular.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a-r-otriangles.jpg\" alt=\"Acute, right, and obtuse triangles\" \/><\/center><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>What is the area of triangle ABC below?<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3abctriangle.jpg\" alt=\"Triangle area question\" \/><\/center><\/p>\n<ol>\n<li>7 in<sup>2<\/sup><\/li>\n<li>13.125 in<sup>2<\/sup><\/li>\n<li>14 in<sup>2<\/sup><\/li>\n<li>14.5 in<sup>2<\/sup><\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice A is correct. The base has a length of 4 in., and the height has a length of 3.5 in., so the area is 7 in<sup>2<\/sup>. If you answered C, you may have forgotten to multiply the product of the base and height by one-half. If you answered D, you may have calculated the perimeter of the triangle.<\/p>\n<p class=\"q-reveal center\"><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/s10_p3_equation.gif\" alt=\"Answer equation\" \/><\/p>\n<\/section>\n<p>Think about why the formula for area contains <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/s10_p3_half.gif\" alt=\"1\/2\" \/>. We\u2019ll address this in a later section.<\/p>\n<h3>Measuring Right Triangles: The Pythagorean Theorem<\/h3>\n<p>This is probably the most popular theorem in all of geometry. In fact, it\u2019s pretty important algebraically, as well.<\/p>\n<p>The right triangle below has legs of length a and b, and a hypotenuse of length c.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3tp-triangle.jpg\" alt=\"Legs and Hypotenuse\" \/><\/center><\/p>\n<p class=\"figcaption\">Legs and Hypotenuse<\/p>\n<p>The Pythagorean Theorem gives the relationship between the lengths of these sides. It says: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. (Note: This is only true for right triangles. If the lengths of the sides of any triangle satisfy the Pythagorean Theorem, the triangle must be a right triangle.)<\/p>\n<p>We can take \u201csquare\u201d in its algebraic and its geometric senses. Algebraically, the Pythagorean Theorem looks like this:<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/p3_2.gif\" alt=\"Pythagorean Theorem\" \/><\/center>The theorem can also be interpreted in the <a class=\"inline cboxElement\" href=\"#geometric\">geometric<\/a> sense.<\/p>\n<div style=\"display: none;\">\n<div id=\"geometric\" style=\"padding: 10px; background: #fff;\">\n<p>In the geometric sense, \u201csquare\u201d is literally a square and the theorem looks like this:<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3pt-geo.jpg\" alt=\"Pythagorean Theorem\" \/><\/center><\/p>\n<p class=\"figcaption\">Pythagorean Theorem<\/p>\n<p>The area of the square with side a is a<sup>2<\/sup>, the area of the square with side b is <sup>b2<\/sup>, and the area of the square with side c is c<sup>2<\/sup>. And the sum of a<sup>2<\/sup> and b<sup>2<\/sup> is c<sup>2<\/sup>. That means that the sum of the areas of the two smaller squares is equal to the area of the largest square.<\/p>\n<\/div>\n<\/div>\n<p>All right, let\u2019s see how to use the theorem. Suppose the two legs of a right triangle measure 3 in. and 4 in. What is the length of the hypotenuse? In the theorem, a and b represent the lengths of the legs, so let a = 3 and b = 4. Now find c:<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/s10_p4_equation.gif\" alt=\"Solving for the hypotenuse\" \/><\/center>The hypotenuse is 5 inches in length.<\/p>\n<p>A 3-4-5 triangle is the most popular Pythagorean triple. A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean Theorem. Multiples of Pythagorean triples are also Pythagorean triples. In other words, since 3-4-5 is a Pythagorean triple, so is 6-8-10 and 9-12-15. Another Pythagorean triple is 5-12-13.<\/p>\n<p>Most, if not all, test questions related to the Pythagorean Theorem involve Pythagorean triples, because they\u2019re easier to compute and they don\u2019t involve irrational numbers (like \u221a2 or 3\u221a5).<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>One leg of a right triangle is 8 cm long and its hypotenuse measures 17 cm. What is the length of the remaining leg?<\/p>\n<ol>\n<li>9 cm<\/li>\n<li>11 cm<\/li>\n<li>15 cm<\/li>\n<li>19 cm<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice C is the correct answer. The Pythagorean Theorem states that a<sup>2<\/sup> + b<sup>2<\/sup> = c<sup>2<\/sup>, where a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse. In this problem, one leg measures 8 cm and the hypotenuse measures 17 cm. So, let a = 8 and c = 17, and find b. The other leg has length 15 cm. Did you figure out that 8-15-17 is also a Pythagorean triple?<\/p>\n<p class=\"q-reveal center\"><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/8.4.a.gif\" alt=\"Answer equation\" \/><\/p>\n<\/section>\n<h3>Measuring Right Triangles: Trigonometric Functions<\/h3>\n<p><strong>Trigonometry<\/strong> literally means \u201ctriangle measure.\u201d The trigonometry (or &#8220;trig&#8221;) that we\u2019ll explore here is restricted to right triangles, so sometimes it\u2019s called <strong>right triangle trigonometry<\/strong>.<\/p>\n<p>A trig function is one that relates the lengths of the sides of a right triangle to one of its angle measures. In this lesson, we\u2019ll explore the three basic trig functions: sine, cosine, and tangent.<\/p>\n<p>The <em>sine<\/em> of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/sin-tri.jpg\" alt=\"Sine\" \/><\/center>The <em>cosine<\/em> of an angle is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3cos-tri.jpg\" alt=\"Cosine\" \/><\/center>The <em>tangent<\/em> of an angle is the ratio of the length of the leg opposite the angle to the side adjacent to the angle.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/tan-tri3.jpg\" alt=\"Tangent\" \/><\/center>Below are a few example problems. While these examples do include solutions using calculators, please note that no calculators will be permitted during the ABCTE certification exam. All of the problems on the certification exam will be able to be solved without the use of a calculator.<\/p>\n<p>Example 1: The base of this right triangle is 10 in. long. What is its height, h?<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3tan-tri2.jpg\" alt=\"Example triangle\" \/><\/center>We\u2019re given an angle measure and the side adjacent to that angle. We want to find the length of the side opposite the given angle. So, we need a trig formula that relates the measure of an angle to the sides opposite and adjacent to that angle. The tangent function does that with the following:<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/8.5.a.gif\" alt=\"Solving for tangent\" \/><\/center>Now, evaluate tan30\u00b0 on a scientific calculator to get tan30\u00b0=0.58.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/8.5.b.gif\" alt=\"Solving for tangent, part 2\" \/><\/center>So the height of the triangle is about 5.8 in.<\/p>\n<p>Example 2: Now let\u2019s find the length of the hypotenuse. We could use the fact that there are 180\u00b0 in a triangle to find the measure of the other acute angle, or we could simply use the angle we\u2019re given. If we do that, we have an angle and the sides opposite and adjacent to it. We want to find the hypotenuse, so we could use either sine or cosine.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/8.5.c.gif\" alt=\"Solving for cosine\" \/><\/center>Now evaluate cos30\u00b0 on a scientific calculator to get cos30\u00b0 = 0.87.<\/p>\n<p>Therefore, the <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/8.5.d.gif\" alt=\"Hypotenuse\" \/>, or about 11.5 in.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Which of the following is the best approximation for leg x in the triangle below? Note that the cos50\u00b0 is .574, the sin50\u00b0 is .766, and the tan50\u00b0 is 1.19.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3tri-question.jpg\" alt=\"Solving for x triangle\" \/><\/center><\/p>\n<ol>\n<li>4 ft.<\/li>\n<li>5.7 ft.<\/li>\n<li>7.9 ft.<\/li>\n<li>12.2 ft.<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice A is the correct answer. We\u2019re given an angle measure and the hypotenuse. We want to find the length of the side adjacent to the given angle, so we need a trig formula that relates the measure of an angle to the adjacent side and to the hypotenuse. The cosine function does that. The value of x is about 4 ft. If you answered B, you may have used the sine function instead of the cosine function.<\/p>\n<p class=\"q-reveal\"><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/8.5.e.gif\" alt=\"Answer equation\" \/><\/p>\n<\/section>\n<h3>Review<\/h3>\n<ul>\n<li>The perimeter of a triangle is the sum of the lengths of the sides of the triangle.<\/li>\n<li>The area of a triangle is given by the formula <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/p3_1.gif\" alt=\"A = 1\/2 bh\" \/>, where b is the base and h is the height, and the base and height are perpendicular.<\/li>\n<li>The Pythagorean Theorem relates the lengths of the sides of a right triangle. It states a<sup>2<\/sup> + b<sup>2<\/sup> = c<sup>2<\/sup>, where a and b are the lengths of the legs and c is the length of the hypotenuse.<\/li>\n<li>Trig functions relate the measure of an angle to the sides of a triangle.<\/li>\n<li>There are three basic trig functions:<br \/>\n<center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/p3_3.gif\" alt=\"Trig functions\" \/> <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/p3_4.gif\" alt=\"Trig functions\" \/> <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/p3_5.gif\" alt=\"Trig functions\" \/><\/center><\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/representational-systems\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/measuring-quadrilaterals\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Measuring Triangles Objective In this next section, we\u2019ll examine some components of a triangle, and review the methods to determine the perimeter and area of triangles. We\u2019ll also refresh your memory about the Pythagorean Theorem (and Pythagorean triples) and delve into some basic trigonometry. Previously Covered Protractors are used [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-263","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/263","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/comments?post=263"}],"version-history":[{"count":11,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/263\/revisions"}],"predecessor-version":[{"id":1395,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/263\/revisions\/1395"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/media?parent=263"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}