Mathematics questions on Triangles

In triangles ABC and DEF, ∠B = ∠E, ∠F = ∠C and AB = 3DE. Then, the two triangles are :

(a) If ∆ABC ∼ ∆PQR in which AB = 6 cm, BC = 4 cm, AC = 8 cm and PR = 6 cm, then find the length of (PQ + QR).

OR

(b) In the given figure, $\frac{\mathrm{QR}}{\mathrm{QS}}=\frac{\mathrm{QT}}{\mathrm{PR}}$ and ∠1 = ∠2, show that ∆PQS ∼ ∆TQR.

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(a) The diagonal BD of parallelogram ABCD intersects the line segment AE at the point F, where E is any point on the side BC. Prove that DF × EF = FB × FA.

OR

(b) In ∆ABC, if AD ⊥ BC and AD² = BD × DC, then prove that ∠BAC = 90°.

Find the length of the median through the vertex B of vertices A(9,  2), B( 3, 7) and C( 1, 10). 

In the given figure, PQ || BC. If AP/PB = 4/13 and AC = 20·4 cm, then the length of AQ is:

Which of the following statements is incorrect?

This section has Very Short Answer (VSA) type question.

(a) In the given figure, D is a point on the side BC of ∆ABC such that ∠ADC = ∠BAC. Show that CA² = CD . CB.

OR

(b) In the given figure, OA . OB = OC . OD. Show that ∠A = ∠C and ∠B = ∠D.

(a) In the given figure, PS/SQ = PT/TR and ∠PST = ∠PRQ. Prove that △PQR is an isosceles triangle.
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OR

(b) In the given figure,  ABE ≅ ACD. Prove that  ADE ~ ABC.

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If in two triangles ∆DEF and ∆PQR, ∠D =∠ Q and ∠R = ∠E, then which of the following is not true?

The measurements of ∆LMN and ∆ABC are shown in the figure given below. The length of side AC is:

In the given figure, in ∆ABC, AD ⊥ BC and ∠BAC = 90°. If BC = 16 cm and DC = 4 cm, then the value of x is:

Long Answer (LA) type question

(a) The sum of the third term and the seventh term of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP.
OR
(b) The minimum age of children eligible to participate in a painting competition is 8 years. It is observed that the age of the youngest boy was 8 years and the ages of the participants, when seated in order of age, have a common difference of 4 months. If the sum of the ages of all the participants is 168 years, find the age of the eldest participant in the painting competition.

△ABC and △PQR are shown in the adjoining figures. The measure of ∠C is:

E and F are points on the sides AB and AC respectively of a △ABC such that AE/EB = AF/FC = 1/2. Which of the following relation is true?

(A) A 1.5 m tall boy is walking away from the base of a lamp post which is 12 m high, at the speed of 2.5 m/sec. Find the length of his shadow after 3 seconds.

OR

(B) In parallelogram ABCD, side AD is produced to a point E and BE intersects CD at F. Prove that △ABE ∼ △CFB

(A) The sides of a right triangle are such that the longest side is 4 m more than the shortest side and the third side is 2 m less than the longest side. Find the length of each side of the triangle. Also, find the difference between the numerical values of the area and the perimeter of the given triangle.

OR

(B) Express the equation $\frac{x-2}{x-3}+\frac{x-4}{x-5}=\frac{10}{3};(x\ne 3,5)$ as a quadratic equation in standard form. Hence, find the roots of the equation so formed.

(A) The corresponding sides of △ABC and △PQR are in the ratio 3 : 5. AD⊥BC and PS⊥QR as shown in the following figures:

(i) Prove that ∆ADC ~ ∆PSR

(ii) If AD = 4 cm, find the length of PS.

(iii) Using (ii) find ar (∆ABC) : ar (∆PQR)

OR

(B) State basic proportionality theorem. Use it to prove the following:
If three parallel lines l, m, n are intersected by transversals q and s as shown in the adjoining figure, then AB/BC = DE/EF.

Which of the following statements is false?

In the adjoining figure, ABCD is a trapezium in which XY || AB || CD. If AX = 2/3 × AD, then CY : YB = 

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AD and PS are median of triangles ABC and PQR respectively such that ∆ABD ∼ ∆PQS. Prove that ∆ABC ∼ ∆PQR.

If a line is drawn to one side of a triangle intersecting the other two sides in distinct points then it divides the two sides in the same ratio. Prove it.

Also, state the converse of the above statement.

AD and PS are angle bisectors of ∠A and ∠P of triangles ABC and PQR. If △ABC ~ △ PQR, prove that △ACD ~ △PRS.

State the converse of basic proportionality theorem.

Also find $\frac{BF}{FC}$ in the following figure, given that AB || DC || EF and $\frac{AE}{ED}=\frac{2}{3}$. Also, find the length of EF if AB = 10 cm and DC = 15 cm.

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In △ABC and △PQR, AD and PS are altitudes such that △ABD ~ △PQS and △ACD ~ △PRS. Prove that △ACD ~ △PRS. Prove that △ABC ~ △PQR.

State the basic proportionality theorem.

Use the theorem to do the following :

In △ABC, AD is the angle bisector of angle A. BA is produced to E such that CE || AD. Prove that $\frac{BD}{DC}=\frac{BA}{AC}$.

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