Question

Case Study Based Base 1. ques 2.1 The Shivani wants to make a wall hanging in the shape of an isosceles triangle for her room as shown in figure. She has cut a piece of cardboard with the given measurements. By looking at the cardboard a few questions rose in her mind, help her find the answers.
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Answer 1

Answer:

90

15

180  

For This Question

Answer 2

Given: from the figure, $AB=5\sqrt{2},AC= 5\sqrt{2}, BC=10, \[\angle ABC = \angle ADC = {90^ \circ }\]$.

Solution:

Part (1.1):

Given:

If D is the midpoint of BC, then find $\[2\angle B\]$.

Understand that, if D is the midpoint of BC, then it will divide BC equally into two parts.

Assume that, BD and DC are x and x respectively.

$\[\begin{array}{l}BC = BD + DC\\ \Rightarrow 10 = x + x\\ \Rightarrow 2x = 10\\ \Rightarrow x = 5\end{array}\]$

$\[ \Rightarrow BD = 5,DC = 5\]$

Consider $\[\Delta ABD\]$,

$\[\begin{array}{l}\cos \theta = \frac{{BD}}{{AB}}\\ \Rightarrow \cos \theta = \frac{5}{{5\sqrt 2 }}\end{array}\]$

$\[\begin{array}{l} \Rightarrow \cos \theta = \frac{1}{{\sqrt 2 }}\\ \Rightarrow \cos \theta = {45^ \circ }\end{array}\]$

Find the value of $\[2\angle B\]$.

$\[\begin{array}{l}\therefore \angle B = {45^ \circ }\\ \Rightarrow 2\angle B = 2 \times {45^ \circ }\\ \Rightarrow 2\angle B = {90^ \circ }\end{array}\]$

Therefore, the value of $\[2\angle B\]$ is $\[{90^ \circ }\]$. Hence, the correct answer is option (b).

Part (1.2):

Find the value of 3AD.

$\[\begin{array}{l}A{B^2} = B{D^2} + A{D^2}\\ \Rightarrow {\left( {5\sqrt 2 } \right)^2} = {5^2} + A{D^2}\\ \Rightarrow A{D^2} = 50 - 25\end{array}\]$

$\[\begin{array}{l} \Rightarrow A{D^2} = 25\\ \Rightarrow AD = 5\end{array}\]$

Therefore,

$\[\begin{array}{l}3AD = 3 \times 5\\ \Rightarrow 3AD = 15\end{array}\]$

Hence, the correct answer is option (a). i.e., 15.

Part (1.3):

Find the value of $\[2\angle DAC\]$.

$\[\begin{array}{l}\angle DAC = \sin \theta \\ \Rightarrow \sin \theta = \frac{{DC}}{{AC}}\\ \Rightarrow \sin \theta = \frac{5}{{5\sqrt 2 }}\end{array}\]$

$\[\begin{array}{l} \Rightarrow \sin \theta = \frac{1}{{\sqrt 2 }}\\ \Rightarrow \sin \theta = {45^ \circ }\\ \Rightarrow \angle DAC = {45^ \circ }\end{array}\]$

Therefore,

$\[\begin{array}{l}2\angle DAC = 2 \times {45^ \circ }\\ \Rightarrow 2\angle DAC = {90^ \circ }\end{array}\]$

Hence, the correct answer is option (b). i.e., $\[{90^ \circ }\]$.