Question

6 multiple choice 2 points
find the area of this triangle.
7 multiple choice 2 points
determine the length of \\( \overline { m n } \\) if \\( \overline { s t } \\) is \\( ( 3 x + 2 ) \\) units.

Explanation:

Step1: Recall the area formula for a right - triangle

The area formula for a right - triangle is \(A=\frac{1}{2}\times base\times height\). In a right - triangle, the two legs can be considered as the base and the height. Here, the legs are \(b = 7\) and \(h=6\).

Step2: Substitute the values into the formula

Substitute \(b = 7\) and \(h = 6\) into the formula \(A=\frac{1}{2}\times b\times h\). We get \(A=\frac{1}{2}\times7\times6\).
First, calculate \(7\times6 = 42\). Then, \(\frac{1}{2}\times42=21\).

Step3: Recall the mid - segment theorem for triangles

The mid - segment theorem states that the length of the mid - segment of a triangle (a segment connecting the mid - points of two sides of a triangle) is half the length of the third side. In triangle \(PMN\), \(ST\) is the mid - segment (since \(S\) and \(T\) are mid - points of \(PM\) and \(PN\) respectively). Let \(MN\) be the third side. According to the mid - segment theorem, \(MN = 2\times ST\).

Step4: Substitute the value of \(ST\)

Given \(ST=(3x + 2)\) units. Then \(MN=2\times(3x + 2)\).
Using the distributive property \(a(b + c)=ab+ac\) (where \(a = 2\), \(b=3x\), \(c = 2\)), we have \(MN=2\times3x+2\times2=6x + 4\) units.

Answer:

1. \(21\)
2. \(6x + 4\)