Write the specified quantity as a function of the specified variable.
-The base of an isosceles triangle is half as long as the two equal sides. Write the area of the triangle as a function of the length of the base.
A) $A = 7 b$
B) $\mathrm { A } = \frac { 15 } { 16 } \mathrm {~b} ^ { 3 }$
C) $A = \frac { \sqrt { 15 } } { 4 } b ^ { 2 }$
D) $A = \frac { 15 } { 16 } b ^ { 2 }$

Question

Quizplus · Accepted Answer

Given that the base of the isosceles triangle is half as long as the two equal sides, let the length of the base be $b$, and the length of the equal sides be $2b$. The height $h$ of the triangle can be found using the Pythagorean theorem in one of the right triangles formed by dropping a perpendicular from the apex to the base: $h = \sqrt{(2b)^2 - \left(\frac{b}{2}\right)^2} = \sqrt{4b^2 - \frac{b^2}{4}} = \sqrt{\frac{16b^2 - b^2}{4}} = \sqrt{\frac{15b^2}{4}} = \frac{\sqrt{15}b}{2}$. The area $A$ of the triangle is $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times b \times \frac{\sqrt{15}b}{2} = \frac{\sqrt{15}}{4}b^2$.

Write the Specified Quantity as a Function of the Specified A=7bA = 7 bA=7b

Write the Specified Quantity as a Function of the Specified $A = 7 b$