Chapter 5: Q5 (page 179)

M, N and T are the midpoints of the sides of $Δ XYZ$ .
If $XY = k$ , then $TN = \underline{?}$

Short Answer

Expert verified

The length of TN is $\frac{k}{2}$ units.

Step by step solution

Step 1. Consider the diagram.

$Δ X Y Z$ is:

Here, M, N and T are midpoints of $Δ X Y Z$ . And $X Y = k$ .

Step 2. Show the calculation.

Consider in triangles TXM and ZXY .

$∠ T X M = ∠ Z X Y$ (Common).

$\frac{T X}{Z X} = \frac{X M}{X Y} = \frac{1}{2}$ (Midpoints).

Therefore, $Δ T X M ~ Δ Z X Y$ (SAS).

$\begin{array}{l} ∠ X M T = ∠ X Y Z \\ ∠ M T X = ∠ Y Z X \end{array}$

Angle XMT and XYZ are the corresponding angles. Hence, line TM and $Z Y$ or ZN are parallel to each other.

Consider in triangles NMY and ZXY .

$∠ N Y M = ∠ Z Y X$ (Common).

$\frac{N Y}{Z Y} = \frac{Y M}{Y X} = \frac{1}{2}$ (Midpoints).

Therefore, $Δ N M Y ~ Δ Z X Y$ (SAS).

$\begin{array}{l} ∠ Y N M = ∠ Y Z X \\ ∠ Y M N = ∠ Y X Z \end{array}$

Angle YNM and YZX are the corresponding angles. Hence, line NM and ZX or ZT are parallel to each other.

Thus, MNZT is parallelogram.

Similarly, TNXM is also a parallelogram

$T N = X M$ .

The side TN is calculated as follows:

$\begin{matrix} T N = X M \\ = \frac{X Y}{2} \\ T N = \frac{k}{2} units \end{matrix}$

Step 3. State the conclusion.

Therefore, the length of TN is $\frac{k}{2}$ units.

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M, N and T are the midpoints of the sides of $Δ XYZ$ .
If $XY = k$ , then $TN = \underline{?}$

Short Answer

Step by step solution

Step 1. Consider the diagram.

Step 2. Show the calculation.

Step 3. State the conclusion.

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M, N and T are the midpoints of the sides of ΔXYZ.If XY=k, thenTN=?¯

Short Answer

Step by step solution

Step 1. Consider the diagram.

Step 2. Show the calculation.

Step 3. State the conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Logic and Functions

Applied Mathematics

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

M, N and T are the midpoints of the sides of $Δ XYZ$ .
If $XY = k$ , then $TN = \underline{?}$