Question

hk is a midsegment of δgij. if ij = z + 41 and hk = -z + 28, what is the value of z? (with a triangle diagram labeled j, k, i, h, g)

Explanation:

Step1: Use the midsegment theorem

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

Step2: Substitute the given expressions

Substitute \( IJ=z + 41 \) and \( HK=-z + 28 \) into \( IJ = 2\times HK \). We get \( z + 41=2(-z + 28) \).

Step3: Expand the right - hand side

Using the distributive property \( a(b + c)=ab+ac \), \( 2(-z + 28)=-2z+56 \). So the equation becomes \( z + 41=-2z + 56 \).

Step4: Solve for \( z \)

Add \( 2z \) to both sides: \( z+2z + 41=-2z+2z + 56 \), which simplifies to \( 3z+41 = 56 \).
Subtract 41 from both sides: \( 3z+41 - 41=56 - 41 \), so \( 3z=15 \).
Divide both sides by 3: \( z=\frac{15}{3}=5 \).

Answer:

\( z = 5 \)