Question

The Segment joining the midpoints of sides PQ and PR (the midsegment) in traingle PQR has length 8x^12x+36. If the length of QR is 4x^2+60x+120, find all possible measures of the midsegment

Answer 1

Answer:

The possible measures of the midsegment are 32 units and 212 units.

Step-by-step explanation:

Given:

A triangle PQR with the midsegment made by sides PQ and PR has length equal to $8x^2+12x+36$ and the base length QR opposite to the midsegment is $4x^2+60x+120$.

From midsegment theorem, we know that, the midsegment is a line parallel to the base opposite to it and half the length of the base length.

Therefore, Midsegment = $\frac{1}{2}\times$ Base length QR

$8x^2+12x+36=\frac{1}{2}(4x^2+60x+120)\\\\8x^2+12x+36=2x^2+30x+60\\\\(8x^2-2x^2)+(12x-30x)+(36-60)=0\\\\6x^2-18x-24=0\\\\6(x^2-3x-4)=0\\\\x^2-3x-4=0\\\\x^2+x-4x-4=0\\\\x(x+1)-4(x+1)=0\\\\(x+1)(x-4)=0\\\\x=-1\ or\ x=4$

Now, midsegment can be calculated using the values of 'x'.

First, plug in -1 for 'x'. This gives,

$Midsegment=8(-1)^2+12(-1)+36\\\\Midsegment=8\times 1-12+36\\\\Midsegment=8-12+36=32\ units$

Now, plug in 4 for 'x'. This gives,

$Midsegment=8(4)^2+12(4)+36\\\\Midsegment=8\times 16-48+36\\\\Midsegment=128+48+36=212\ units$

Therefore, the possible measures of the midsegment are 32 units and 212 units