CBSE · NCERT · Class 7 Maths · Chapter 6

NCERT Solutions: Class 7 Maths Chapter 6 - The Triangle and its Properties

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Chapter-wise NCERT intext questions and exercise answers for The Triangle and its Properties, grounded in the official textbook.

Questions are taken verbatim from the NCERT textbook; answers were grounded against the chapter's content during generation. Items needing review are marked.
Sections in this chapter
Exercise 6.1 1Exercise 6.2 1Exercise 6.3 1Exercise 6.4 1Exercise 6.5 1
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1Exercise 6.11 questions
Q.11. In $\triangle PQR$, D is the mid-point of $\overline{QR}$. The figure shows P joined to Q and R, point M on $QR$ with $PM\perp QR$, and point D on $QR$ with $QD=DR$. $\overline{PM}$ is _________________. $\overline{PD}$ is _________________. Is $QM=MR$? 2. Draw rough sketches for the following: (a) In $\triangle ABC$, BE is a median. (b) In $\triangle PQR$, PQ and PR are altitudes of the triangle. (c) In $\triangle XYZ$, YL is an altitude in the exterior of the triangle. 3. Verify by drawing a diagram if the median and altitude of an isosceles triangle can be same.v
Solution

An altitude is perpendicular to the opposite side or its extension, while a median joins a vertex to the midpoint of the opposite side.

Answer:

1. $\overline{PM}$ is an altitude. $\overline{PD}$ is a median. No, $QM$ need not be equal to $MR$.
2. (a) Draw $\triangle ABC$ and mark E as the midpoint of AC; join B to E. (b) Draw a right-angled $\triangle PQR$ with right angle at P; then PQ is perpendicular to PR and PR is perpendicular to PQ, so PQ and PR are altitudes. (c) Draw an obtuse $\triangle XYZ$ and extend side XZ; draw YL perpendicular to the extended line XZ outside the triangle.
3. Yes. In an isosceles triangle, the line from the vertex angle to the midpoint of the base is also perpendicular to the base; hence the median and altitude can be the same.

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2Exercise 6.21 questions
Q.1-21. Find the value of the unknown exterior angle $x$ in the following diagrams: (i) the two interior opposite angles are $50^\circ$ and $70^\circ$; (ii) the two interior opposite angles are $65^\circ$ and $45^\circ$; (iii) the two interior opposite angles are $30^\circ$ and $40^\circ$; (iv) the two interior opposite angles are $60^\circ$ and $60^\circ$; (v) the two interior opposite angles are $50^\circ$ and $50^\circ$; (vi) the two interior opposite angles are $30^\circ$ and $60^\circ$. 2. Find the value of the unknown interior angle $x$ in the following figures: (i) exterior angle $115^\circ$ and other interior opposite angle $50^\circ$; (ii) exterior angle $100^\circ$ and other interior opposite angle $70^\circ$; (iii) exterior angle $125^\circ$ and the other marked angle is a right angle; (iv) exterior angle $120^\circ$ and other interior opposite angle $60^\circ$; (v) exterior angle $80^\circ$ and other interior opposite angle $30^\circ$; (vi) exterior angle $75^\circ$ and other interior opposite angle $35^\circ$.v
Solution

Use the exterior angle property: an exterior angle of a triangle is equal to the sum of its two interior opposite angles.

Answer:

1. (i) $120^\circ$ (ii) $110^\circ$ (iii) $70^\circ$ (iv) $120^\circ$ (v) $100^\circ$ (vi) $90^\circ$.
2. (i) $65^\circ$ (ii) $30^\circ$ (iii) $35^\circ$ (iv) $60^\circ$ (v) $50^\circ$ (vi) $40^\circ$.

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3Exercise 6.31 questions
Q.1-21. Find the value of the unknown $x$ in the following diagrams: (i) a triangle has angles $50^\circ$, $60^\circ$ and $x$; (ii) a right triangle has one right angle, a $30^\circ$ angle and $x$; (iii) a triangle has angles $30^\circ$, $110^\circ$ and $x$; (iv) a triangle has one angle $50^\circ$ and the other two angles both marked $x$; (v) all three angles are marked $x$; (vi) a right triangle has one right angle and the other two angles marked $x$ and $2x$. 2. Find the values of the unknowns $x$ and $y$ in the following diagrams: (i) a triangle has one angle $50^\circ$, an exterior angle $120^\circ$ at the base, the interior adjacent angle $y$, and top angle $x$; (ii) a triangle has left base angle $50^\circ$, right base angle $x$, top interior angle $y$, and the vertically opposite exterior angle above the top marked $80^\circ$; (iii) a triangle has angles $50^\circ$, $60^\circ$, and interior base angle $y$, with exterior angle $x$ adjacent to $y$; (iv) a triangle has left base angle $30^\circ$, right base angle $x$, top angle $y$, and an exterior angle $60^\circ$ adjacent to $x$; (v) a triangle has base angles both $x$, top interior angle $y$, and the vertically opposite exterior angle at the top marked $90^\circ$; (vi) a triangle is formed by two slant lines meeting at the top and a horizontal line, with exterior angles $x$ at the top and at both base extensions, and the interior right base angle $y$.v
Solution

Use $\angle A+\angle B+\angle C=180^\circ$ for every triangle, along with linear pair and equal-angle information shown in each figure.

Answer:

1. (i) $70^\circ$ (ii) $60^\circ$ (iii) $40^\circ$ (iv) $65^\circ$ (v) $60^\circ$ (vi) $30^\circ$.
2. (i) $x=70^\circ$, $y=60^\circ$ (ii) $x=50^\circ$, $y=80^\circ$ (iii) $x=110^\circ$, $y=70^\circ$ (iv) $x=60^\circ$, $y=90^\circ$ (v) $x=45^\circ$, $y=90^\circ$ (vi) $x=60^\circ$, $y=60^\circ$.

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4Exercise 6.41 questions
Q.1-6Exercise 6.4 asks whether triangles can be constructed from given side lengths and uses the triangle inequality property.v
Solution

In a triangle, the sum of any two sides must be greater than the third side. If two sides are 15 and 12, the third side must be greater than $15-12=3$ and less than $15+12=27$.

Answer:

1. (i) Not possible (ii) Possible (iii) Not possible.
2. (i) Yes (ii) Yes (iii) Yes.
3. Yes.
4. Yes.
5. Yes.
6. The third side must be between 3 and 27.

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5Exercise 6.51 questions
Q.1-81. PQR is a triangle, right-angled at P. If $PQ=10$ cm and $PR=24$ cm, find QR. 2. ABC is a triangle, right-angled at C. If $AB=25$ cm and $AC=7$ cm, find BC. 3. A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance $a$. Find the distance of the foot of the ladder from the wall. 4. Which of the following can be the sides of a right triangle? (i) 2.5 cm, 6.5 cm, 6 cm. (ii) 2 cm, 2 cm, 5 cm. (iii) 1.5 cm, 2 cm, 2.5 cm. In the case of right-angled triangles, identify the right angles. 5. A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base of the tree. Find the original height of the tree. 6. Angles Q and R of a $\triangle PQR$ are $25^\circ$ and $65^\circ$. Write which of the following is true: (i) $PQ^2+QR^2=RP^2$ (ii) $PQ^2+RP^2=QR^2$ (iii) $RP^2+QR^2=PQ^2$. 7. Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm. 8. The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.v
Solution

For a right triangle, $\text{hypotenuse}^2=\text{leg}_1^2+\text{leg}_2^2$. To check whether a triangle is right-angled, square the longest side and compare it with the sum of squares of the other two sides.

Answer:

1. 26 cm.
2. 24 cm.
3. 9 m.
4. (i) and (iii).
5. 18 m.
6. (ii).
7. 98 cm.
8. 68 cm.

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