Question

similar figures test (topic 5)
if △hbn ~ △lyr, find the value of x
b
28
h 3x - 7 n
r
x + 11
y 21 l
x=

Explanation:

Step1: Set up proportion for similar - triangles

For similar triangles $\triangle BHN$ and $\triangle LYR$, the ratios of corresponding sides are equal. So, $\frac{28}{x + 11}=\frac{3x - 7}{21}$.

Step2: Cross - multiply

Cross - multiplying gives us $28\times21=(3x - 7)(x + 11)$.
$588 = 3x^{2}+33x-7x - 77$.

Step3: Simplify the equation

$3x^{2}+26x-77 - 588=0$.
$3x^{2}+26x - 665 = 0$.

Step4: Solve the quadratic equation

We can use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$. Here, $a = 3$, $b = 26$, and $c=-665$.
First, calculate the discriminant $\Delta=b^{2}-4ac=(26)^{2}-4\times3\times(-665)=676 + 7980=8656$.
Then, $x=\frac{-26\pm\sqrt{8656}}{6}=\frac{-26\pm93.03}{6}$.
We get two solutions:
$x_1=\frac{-26 + 93.03}{6}=\frac{67.03}{6}\approx11.17$ and $x_2=\frac{-26 - 93.03}{6}=\frac{-119.03}{6}\approx - 19.84$.
Since the length of a side cannot be negative in this context, we discard the negative solution.

Answer:

$x = 11$ (rounded to the nearest whole number)