Question

what is hk? hk =

Explanation:

Step1: Identify congruent triangles

Since $\angle HIK=\angle JIK = 40^{\circ}$, $\angle IHK=\angle IJK = 90^{\circ}$ and $IK$ is common to both $\triangle HIK$ and $\triangle JIK$, by the Angle - Angle - Side (AAS) congruence criterion, $\triangle HIK\cong\triangle JIK$.

Step2: Set up an equation using congruent sides

Corresponding sides of congruent triangles are equal. So, $HK = JK$. We have $HK=8p + 8$ and $JK=p + 43$. Then $8p+8=p + 43$.

Step3: Solve the equation for $p$

Subtract $p$ from both sides: $8p - p+8=p - p + 43$, which simplifies to $7p+8 = 43$. Then subtract 8 from both sides: $7p+8 - 8=43 - 8$, giving $7p=35$. Divide both sides by 7: $p=\frac{35}{7}=5$.

Step4: Find the length of $HK$

Substitute $p = 5$ into the expression for $HK$. $HK=8p + 8=8\times5+8=40 + 8=48$.

Answer:

48