SSC (English Medium) १० वीं कक्षा - Maharashtra State Board Question Bank Solutions for Geometry Mathematics 2

Show that A(1, 2), (1, 6), C(1 + 2 `sqrt(3)`, 4) are vertices of a equilateral triangle

Seg OA is the radius of a circle with centre O. The coordinates of point A is (0, 2) then decide whether the point B(1, 2) is on the circle?

Using distance formula decide whether the points (4, 3), (5, 1), and (1, 9) are collinear or not.

If a and b are natural numbers and a > b If (a² + b²), (a² – b²) and 2ab are the sides of the triangle, then prove that the triangle is right-angled. Find out two Pythagorean triplets by taking suitable values of a and b.

Prove that, The areas of two triangles with the same height are in proportion to their corresponding bases. To prove this theorem start as follows:

1. Draw two triangles, give the names of all points, and show heights.
2. Write 'Given' and 'To prove' from the figure drawn.

Find distance between points P(– 5, – 7) and Q(0, 3).

By distance formula,

PQ = `sqrt(square + (y_2 - y_1)^2`

= `sqrt(square + square)`

= `sqrt(square + square)`

= `sqrt(square + square)`

= `sqrt(125)`

= `5sqrt(5)`

If m and n are real numbers and m > n, if m² + n², m² – n² and 2 mn are the sides of the triangle, then prove that the triangle is right-angled. (Use the converse of the Pythagoras theorem). Find out two Pythagorian triplets using convenient values of m and n.

If ΔABC ∼ ΔDEF, length of side AB is 9 cm and length of side DE is 12 cm, then find the ratio of their corresponding areas.

What is the distance of the point (– 5, 4) from the origin?

In the figure, PQ ⊥ BC, AD ⊥ BC. To find the ratio of A(ΔPQB) and A(ΔPBC), complete the following activity.

Given: PQ ⊥ BC, AD ⊥ BC

Now, A(ΔPQB)  = `1/2 xx square xx square`

A(ΔPBC)  = `1/2 xx square xx square`

Therefore, 

`(A(ΔPQB))/(A(ΔPBC)) = (1/2 xx square xx square)/(1/2 xx square xx square)`

= `square/square`

Find the value of y, if the points A(3, 4), B(6, y) and C(7, 8) are collinear.

Draw a line segment AB of length 10 cm and divide it internally in the ratio of 2:5 Justify the division of line segment AB.

Find the distance between the points O(0, 0) and P(3, 4).

Show that points A(–1, –1), B(0, 1), C(1, 3) are collinear.

In the following figure, in ΔABC, ∠B = 90°, ∠C = 60°, ∠A = 30°, AC = 18 cm. Find BC.

Find the height of an equilateral triangle whose side is 6 units.

In the given figure. Find RP and PS using the information given in ∆PSR.

In the given figure, ∆PQR is an equilateral triangle. Point S is on seg QR such that QS = n\[\frac{1}{3}\] QR.

Prove that: 9 PS² = 7 PQ²

In right-angled triangle PQR, if ∠P = 60°, ∠R = 30° and PR = 12, then find the values of PQ and QR. 

In ΔDEF, if ∠E = 90°, then find the value of ∠D + ∠F.