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Chapter 11: Q. 90 (page 774)

Geometry: Find formulas for the base b and one of the equal sides l of an isosceles triangle in terms of its altitude h and perimeter P.

Short Answer

Expert verified

We have $l = \frac{4 h^{2} + P^{2}}{4 P}$ and $b = \frac{P^{2} - 4 h^{2}}{2 P}$ .

Step by step solution

01

Step 1. Given information

For an isosceles triangle,

$h = \frac{\sqrt{4 l^{2} - b^{2}}}{2}$

and $P = 2 l + b$ .

02

Step 2. Solve the system of equations.

From second equation,

$P = 2 l + b P - 2 l = b b = P - 2 l$

Substitute this in first equation,

role="math" localid="1646984519275" $h = \frac{\sqrt{4 l^{2} - {(P - 2 l)}^{2}}}{2} h^{2} = \frac{4 l^{2} - (P^{2} - 4 P l + 4 l^{2})}{4} 4 h^{2} = 4 l^{2} - P^{2} + 4 P l - 4 l^{2} 4 h^{2} = 4 P l - P^{2} 4 h^{2} + P^{2} = 4 P l \frac{4 h^{2} + P^{2}}{4 P} = l$

03

Step 3. Substitute l=4h2+P24P in b=P-2l.

We get

$b = P - \frac{4 h^{2} + P^{2}}{4 P} \times 2 = P - \frac{4 h^{2} + P^{2}}{2 P} = \frac{P^{2} - 4 h^{2}}{2 P}$

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Most popular questions from this chapter

The ratio of a to b is $\frac{2}{3}$ . The sum of a and b is $10$ . What is the ratio of $a + b$ to $b - a$ ?

Solve each system of equations. If the system has no solution, say that it is inconsistent.
$\{\begin{matrix} x + 3 y = 5 \\ 2 x - 3 y = - 8 \end{matrix}$

Graph the equation: $y = 3 x + 2$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$x - y - z = 1 2 x + 3 y + z = 2 3 x + 2 y = 0$

In Problems 35–42, graph each system of inequalities by hand.

$\{\begin{cases} x^{2} + y^{2} \leq 9 \\ x + y \geq 3 \end{cases}$

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