  # Math Question

<p>What is the greatest possible area of a triangle with one side of length 7 and another side of length 10?</p>

<p>a. 17
b. 34
c. 35
d. 70
e. 140</p>

<p>The answer is C but i put d.</p>

<p>1/2<em>b</em>h
1/2<em>7</em>10=35</p>

<p>The sides are 7 and 10. We know that for all legs of a triangle the other sum of the other two must be greater than the third. since 7+10=17 the longest side, or the hypotenuse, is 16. However, the hypotenuse is not used in calculating area. We use 1/2(b<em>h) to find area thus 7</em>10=70*.5=35</p>

<p>Let's see: 7 and 10 be the short legs and hypotenuse is unknown. So it's 70/2 = 35.</p>

<p>If 10 was to be the hypotenuse, largest length for the base can be 9.</p>

<p>7 (9) = 63. 63 < 70.</p>

<p>Therefore, biggest area is 35.</p>

<p>The three posts above mine take for granted that the triangle is a right triangle, even though the question doesn't say that it's a right triangle.</p>

<p>It's a right triangle because Area = (1/2) (Base) (Height) no matter what the angles of the triangle are. For example, the area of a right triangle with legs of lengths 4 units and 5 units has an area of 10 units^2. If you put the 5-unit-long leg on the bottom, put the vertex between the hypotenuse and the 4-unit-long leg in the upper-left-hand corner, and stretch that vertex 100 units to the left, then the triangle still has an area of 10 square units.</p>

<p>So, since the base doesn't change but you determine the height, what's the maximum height that the triangle can possibly have? That maximum height is the length of the other leg.</p>

<p>The length of the other leg can't be the max height because it isn't vertical (i.e. you need to find the distance from vertex straight down to the opposite leg and use that number as the height). The right triangle does have the greatest area.</p>

<p>With one leg of a right triangle on the bottom, the other leg is vertical.</p>

<p>Here's another way to picture it:</p>

<p>Suppose the two lengths were attached with a hinge. Now lay one of the sides down on the ground and call it the base. As you change the angle of the hinge, the base stays the same, but the height gets taller as the angle at the hinge gets closer to 90 degrees. And the maximum height is when the hinge is exactly perpendicular.</p>

<p>And here's another way, but it requires trig:</p>

<p>There is a formula that gives the area of a triangle as a function of the lengths of the two sides and the angle between them.</p>

<p>A= (1/2) x a x b x sin(c).</p>

<p>In this problem, a and b are fixed as 7 and 10. And the largest possible value for sine of an angle is 1. (That occurs when the angle is 90 but you don't have to know that to get the problem right.) So max area is (1/2)<em>7</em>10</p>