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PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM^2 = QM × MR.

Given: ΔPQR is right-angled at P is a point on QR such that PM ⊥QR. To Prove: PM2 = QM × MR Proof: ΔPMQ and ΔPMR are right triangles... View Article

D and E are points on sides AB and AC respectively of ΔABC such that ar(DBC) = ar(EBC) . Prove that DE || BC.

Solution ΔDBC and ΔEBC are on the same base BC and are having equal areas. Hence, They will lie between the same parallel... View Article

The height of a cone is 15cm. If its volume is 1570cm³, find the diameter of its base. (Use π = 3.14)

Given Height of the cone, h = 15 cm The volume of cone =1570 cm3 Find out We have to find out the diameter of its base Solution Volume of... View Article

The diameter of a sphere is decreased by 25%. By what percent does its curved surface area decrease?

Given The diameter of a sphere is decreased by 25% Find out We have to find out by what per cent does its curved surface area decreases?... View Article

In the figure, ABC and AMP are two right triangles, right angled at B and M respectively, prove that:

Given ABC and AMP are two right triangles, right angled at B and M respectively To Prove (i) ΔABC ~ ΔAMP (ii) CA/PA = BC/MP Proof... View Article

What is a regular polygon? State the name of a regular polygon of (i) 3 sides (ii) 4 sides (iii) 6 sides

A polygon is a closed two-dimensional figure, which is made up of straight lines. Whenall the sides and interior angles of the polygon are... View Article

Give the geometric representations of 2x + 9 = 0 as an equation (i) in one variable (ii) in two variables

Firstly, we solve 2x + 9 = 0, to get x = -9/2 (i) If x = -9/2 is treated as an equation in one variable x only, then it has the unique... View Article

Find the cost of digging a cuboidal pit 8m long, 6m broad and 3m deep at the rate of Rs 30 per m³.

Given Length of the cuboidal pit, l = 8 m Width of the cuboidal pit, b = 6m Height of cuboidal pit, h = 3m Find out We have to... View Article

In Fig. ar(DRC) = ar(DPC) and ar(BDP) = ar(ARC) . Show that both the quadrilaterals ABCD and DCPR are trapeziums.

Solution Given ar(ΔDRC) = ar(ΔDPC) ar(ΔBDP) = ar(ΔARC) Prove: ABCD and DCPR are trapeziums. Proof:... View Article

The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.

Literacy rate (in %) 45-55 55-65 65-75 75-85 85-98 Number of cities 3 10 11 8 3 Solution Class Interval Frequency (fi) (xi)... View Article

List the outcomes you can see in these experiments. (a) Spinning a wheel (b) Tossing two coins together

Solution (a) Since, there are four letters A, B, C and D in a spinning wheel Therefore there are 4 possible outcomes. (b) When two coins are... View Article

In , sides AB and AC of ΔABC are extended to points P and Q respectively. Also, PBC < QCB. Show that AC > AB.

Solution Given ∠PBC < ∠QCB To Prove PBC < QCB We know that ABC + PBC = 180° So, ABC = 180°-PBC Also, ACB +QCB = 180° Thus, ACB =... View Article

2 cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid.

Volume of cube=64cm3 = (side)3 (Side)3 = 64 Side = (64cm3) 1/3 = (2*2*2*2*2*2 cm3)1/3 = 2*2 = 4 cm Hence, Length = 8... View Article

If Q (0, 1) is equidistant from P (5, – 3) and R (x, 6) , find the values of x. Also find the distance QR and PR.

Given: Q (0, 1) is equidistant from P (5, – 3) and R (x, 6), which means PQ = QR Let us find the distance between PQ and QR using distance... View Article

If AD and PM are medians of triangles ABC and PQR, respectively where ΔABC ~ ΔPQR prove that AB/PQ = AD/PM.

Given that ΔABC ~ ΔPQR To Prove AB/PQ = AD/PM Proof We know that the corresponding sides of similar triangles are in proportion. So,... View Article

In Figure, D is a point on side BC of ∆ ABC such that BD/CD = AB/AC. Prove that AD is the bisector of ∠ BAC.

In the given figure, let us extend BA to P such that; AP = AC. Now join PC. In ΔAEC, AE=AC∠AEC=∠ACE From figure AC=AE. BDCD=ABAE... View Article

Construct the following quadrilateral: Quadrilateral GOLD. OL = 7.5 cm GL = 6 cm GD = 6 cm LD = 5 cm OD = 10 cm

Solution The rough sketch of the quadrilateral GOLD can be drawn as follows. (1) ∆ GDL can be constructed by using the given measurements as... View Article

Find the measure of ∠P and ∠S if SP || RQ ? in Fig (If you find m∠R, is there more than one method to find m∠P?)

∠P + ∠Q = 180° {angles on the same side of transversal} ⇒ ∠P + 130° = 180° ⇒ ∠P = 180° –... View Article

Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Given Given that, OA = OC OB = OD and ∠AOB = ∠BOC = ∠OCD = ∠ODA = 90° AC = BD To Prove If the diagonals of a quadrilateral bisect each... View Article

If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

Given that S7 = 49 S17 = 289 Find out We need to find the sum of first n terms Solution We know that Sum of n terms;... View Article

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