Question

if hi = hk = 47, ij = p + 42, and jk = 3p, what is jk?

Explanation:

Step1: Identify congruent triangles

Since $HI = HK$ and $\angle IHJ=\angle KHJ = 90^{\circ}$ and $HJ$ is common to both right - triangles $\triangle IHJ$ and $\triangle KHJ$, by the Hypotenuse - Leg (HL) congruence theorem, $\triangle IHJ\cong\triangle KHJ$.

Step2: Set up an equation for side lengths

Because the two triangles are congruent, $IJ = JK$. Given $IJ=p + 42$ and $JK = 3p$, we can set up the equation $p + 42=3p$.

Step3: Solve the equation for $p$

Subtract $p$ from both sides: $42=3p - p$, which simplifies to $42 = 2p$. Then divide both sides by 2: $p=\frac{42}{2}=21$.

Step4: Find the length of $JK$

Since $JK = 3p$, substitute $p = 21$ into the expression. So $JK=3\times21 = 63$.

Answer:

63