Question

2 you need compass, straightedge and paper.
a) draw segment ab.
b) draw the perpendicular bisector of ab.
c) on the perpendicular line, mark - off point p such that the distance from p to segment ab is greater than 0.5ab and less than segment ab.
d) what is name of the triangle abp?
a) what can you say about side ap and bp?
2 look at the given figure.
a) what is the value of x?
b) what is name of triangle klm? why do you say so?

Explanation:

Step1: Analyze triangle ABP

Since P lies on the perpendicular bisector of AB, by the property of the perpendicular bisector, any point on the perpendicular bisector of a line - segment is equidistant from the endpoints of the line - segment. So, AP = BP. A triangle with two equal sides is an isosceles triangle. So, triangle ABP is an isosceles triangle.

Step2: Analyze triangle KLM

In triangle KLM, if KM = LM (assuming the lengths represented by the expressions are equal for the sides), we set up the equation 3x + 2=x + 6.
Subtract x from both sides: 3x - x+2=x - x + 6, which simplifies to 2x+2 = 6.
Subtract 2 from both sides: 2x+2 - 2=6 - 2, getting 2x = 4.
Divide both sides by 2: $\frac{2x}{2}=\frac{4}{2}$, so x = 2.
Since KM = LM (from the equation solving), triangle KLM is an isosceles triangle because it has two equal sides.

Answer:

1. d) Isosceles triangle

a) AP = BP

1. a) x = 2

b) Isosceles triangle. Because KM = LM (by solving 3x + 2=x + 6 for x and substituting to show the side - lengths are equal).