Question

gi is a midsegment of \\( \triangle hjk \\).
if \\( jk = 6p + 34 \\) and \\( gi = 7p - 23 \\), what is the value of \\( p \\)?

Explanation:

Step1: Apply the midsegment theorem

The midsegment theorem states that the length of a midsegment of a triangle ($GI$) is half the length of the parallel side ($JK$). So, $JK = 2\times GI$.

Step2: Substitute the given expressions

Substitute $JK = 6p + 34$ and $GI = 7p - 23$ into $JK = 2\times GI$. We get $6p + 34=2(7p - 23)$.

Step3: Expand the right - hand side

Using the distributive property $a(b + c)=ab+ac$, where $a = 2$, $b = 7p$, and $c=-23$. So, $6p + 34 = 14p-46$.

Step4: Move the variable terms to one side

Subtract $6p$ from both sides: $34=14p - 6p-46$.

Step5: Simplify the variable terms

$34 = 8p-46$.

Step6: Move the constant terms to one side

Add $46$ to both sides: $34 + 46=8p$.

Step7: Simplify the left - hand side

$80 = 8p$.

Step8: Solve for $p$

Divide both sides by $8$: $p=\frac{80}{8}$.

Answer:

$10$