Question

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Please help me with #8 and 9

Answer 1

8. TN = 22 units

9. RT = √(165) units

Explanation:

It appears that point T is the circumcenter of triangle MNP.

A circumcenter of a triangle is:

• The center of a circle that passes through each vertex of a triangle.
• The point at which the perpendicular bisectors of the sides of the triangle intersect.

As a perpendicular bisector is a line that divides another line segment into two equal parts at a right angle, then:

• MR = RN
• NS = SP
• PQ = QM

As point T is the circumcenter, ΔMTN, ΔNTP and ΔMTP are isosceles triangles and MT, TP and TN are the radius of the circumcircle.

Therefore:

• MT = TP = TN

Question 8

To find the length of TN, find the value of x by equating the given expressions for MP and TP:

`\implies \sf MT = TP`

`\implies 6x - 56 = 3x - 17`

`\implies 3x = 39`

`\implies x = 13`

Substitute the found value of x into the expression for TP:

`\implies \textsf{TP} = 3(13) - 17`

`\implies \textsf{TP} = 22`

Therefore, as TN = TP then TN = 22 units.

Question 9

First find the value of x by equating the given expressions for TN and TP:

`\implies \sf TN=TP`

`\implies 4x-17=x+10`

`\implies 3x=27`

`\implies x=9`

Substitute the found value of x into the expression for TN:

`\implies \textsf{TN} = 4(9) - 17`

`\implies \textsf{TN} = 19`

As RT is the perpendicular bisector of MN, then ΔTRN is a right triangle where ∠TRN is 90°.

Given:

• RN = 14
• TN = 19

To calculate the length of RT, use Pythagoras Theorem:

`\implies \sf RN^2+RT^2=TN^2`

`\implies \sf RT=√(TN^2-RN^2)`

`\implies \sf RT=√(19^2-14^2)`

`\implies \sf RT=√(165)`