Construction of Triangles: When 2 Angles and Included Side Are Given

Q.

In Fig. the line segment joining the mid-points M and N of opposite sides AB and DC of quadrilateral ABCD is perpendicular to both these sides. Prove that the other sides of the quadrilateral are equal. [4 MARKS]

Q. Which of the following informations are applicable for constructing a triangle using ASA criteria?

1. $△$ ABC , with $\angle$ A = 75°, $\angle$ B = 55° and $\overline{AB}$ =4 cm
2. $△$ EFG , with $\angle$ E = 80°, $\angle$ G = 25° and $\overline{EF}$ =7 cm
3. $△$ XYZ , right angled at $\angle$ X , $\angle$ Y = 40° and $\overline{XY}$ =12 cm
4. Isosceles triangle $△$ PQR , with base $\overline{PQ}$ = 5 cm and $\angle$ P = 65°

Q. In $△$ PQR , $\angle$ Q = 90° and $\overline{PQ}$ = 15 cm. Which of the following information is needed to construct the triangle?

1. $\overline{QR}$ = 8 cm.
2. $\overline{PR}$ = 17 cm.
3. $\angle$ P = 30°
4. $\angle$ R = 60°

Q. In $△$ PQR , $\angle$ P = 75° and $\overline{PQ}$ =8 cm. Select the correct information needed to construct the triangle?

1. $\angle$ Q = 75°
2. $\angle$ R = 30°
3. $\overline{QR}$ = 12 cm
4. $\overline{PR}$ = 12 cm

Q. In $△$ DEF , $\angle$ D = 55° and $\angle$ E = 105°. Which is the correct information needed to construct the triangle?

1. $\angle$ F = 20°
2. $\overline{DE}$ =3.5 cm
3. $\overline{EF}$ =6.5 cm
4. $\overline{DF}$ =11 cm

Q.

In an isosceles $\Delta$ABC, $\angle$A is 30${}^{\circ }$. The sides AB and AC are equal. Using constructions, find which side of $\Delta$ABC is the shortest.

1. AB

2. Data Insufficient

3. BC

4. AC

Q. Steps for the construction of a triangle XYZ, in which XY = 4.5 cm, $\angle X={100}^{\circ }$ and $\angle Y={50}^{\circ }$ is given below. Choose the correct order.

1. Construct a line segment XP such that $\angle PXY={100}^{\circ }$
2. Locate points X and Y on it such that XY = 4.5 cm
3. Draw a line segment which is sufficiently long using ruler
4. Extend XP and YQ to intersect at Z.
5. Construct a line segment YQ such that $\angle XYQ={50}^{\circ }$

1. 3, 2, 5, 1, 4
2. 2, 3, 1, 5, 4
3. 3, 2, 4, 1, 5
4. 2, 1, 3, 4, 5

Q. The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides

1. True
2. False

Q.

In an isosceles $\Delta$ABC, $\angle$A is 30${}^{\circ }$. The sides AB and AC are equal. Using constructions, find which side of $\Delta$ABC is the shortest.

1. BC

2. AC

3. AB

4. Data Insufficient

Q. Steps for the construction of an equilateral triangle of height 3.2cm is given below. Choose the correct order.

1.Draw a line segment XY
2.Take any point M on XY. Draw ZM$\perp$XY
3.With M as centre and radius 3.2cm , draw an arc , cutting MZ at A.
4.Construct $\angle MAB={30}^{\circ }$and $\angle MAC={30}^{\circ }$ , with B and C on XY.

1. 2, 1, 3, 4
2. 1, 2, 4, 3
3. 1, 2, 3, 4
4. 2, 1, 4, 3

Q. In $△$ XYZ , $\overline{XY}$ =6.3 cm and $\overline{XZ}$ = 7 cm. Which is the correct information needed to construct the triangle?

1. $\angle$ X = 85°
2. $\angle$ Y = 50°
3. None of the above
4. $\angle$ Z = 45°

Q. The length of the hypotenuse of a right-angle triangle whose measure of two sides are 12 cm and 0.35 m is:

1. 37 cm
2. 3.72 cm
3. 0.372 cm
4. 37 m

Q.

$PQR$ is a triangle, right angled at $P$. If $PQ=10cm$ and $PR=24cm$, find $QR$.

Q. For construction of a $△PQR$, where $QR=8cm,PR=10cm$ and $\angle Q={90}^{\circ }$, its steps for construction is given below in jumbled form. Identify the first step from the following.

1. 1
2. 2
3. 3
4. 4

Q.

In an isosceles $\Delta ABC$, $\angle A$ is ${30}^{\circ }$. The sides AB and AC are equal. Using constructions, find which side of $\Delta ABC$ is the shortest.

1. BC

2. AC

3. Data Insufficient

4. AB

Q. Steps for the construction of a triangle ABC whose perimeter is 12.5 cm and whose base angles are ${60}^{\circ }$ and ${75}^{\circ }$ are given below. Choose the correct order.

1.Join AB and AC
2.Draw a line segment PQ of length 12.5cm
3.Construct rays PR such that $\angle QPR={60}^{\circ }$ and QS such that $\angle PQS={75}^{\circ }$
4.Draw the perpendicular bisector of AP and AQ and let these intersect PQ at B and C respectively.
5.Draw the bisectors PL and QM of $\angle QPR$ and $\angle PQS$ respectively. Let these intersect at A.

1. 2, 3, 1, 5, 4
2. 2, 3, 4, 1, 5
3. 2, 3, 1, 5, 4
4. 2, 3, 5, 4, 1

Q. Steps for the construction of a isosceles triangle ABC whose altitude is 4cm and vertex angle is ${80}^{\circ }$ is given below. Choose the correct order.

1.With M as centre and radius 4cm, draw an arc cutting MP at A.
2.Construct B and C on XY such that $\angle MAB=\frac{80}{2}={40}^{\circ }$ and $\angle MAC=\frac{80}{2}={40}^{\circ }$
3.Take a point M on point XY, and draw a line MP$\perp$ XY
4.Draw a line segment XY.

1. 4, 3, 2, 1
2. 4, 2, 1, 3
3. 1, 3, 2, 4
4. 4, 3, 1, 2

Q. In $\Delta ABC;m\angle B={90}^{\circ };AB=5cm$ and $AC=13$ then, $BC=....$

1. $10$
2. $8$
3. $12$
4. $18$

Q.

In an isosceles $\Delta$ABC, $\angle$A is 30${}^{\circ }$. The sides AB and AC are equal. Using constructions, find which side of $\Delta$ABC is the shortest.

1. BC

2. AC

3. AB

4. Data Insufficient