`triangleDEF ~ triangleMNK`. If DE = 5 and MN = 6, then find the value of `(A(triangleDEF))/(A(triangleMNK))`

Concept: Similarity of Triangles

If two circles with radii 8 cm and 3 cm respectively touch externally, then find the distance between their centres.

Concept: Touching Circles

Find the length of the altitude of an equilateral triangle with side 6 cm.

Concept: Construction of Triangle If the Base, Angle Opposite to It and Either Median Altitude is Given

If `theta = 45^@`, then find `tan theta`.

Concept: Angles in Standard Position

A slope of a line is 3 and y-intercept is –4. Write the equation of a line

Concept: Slope of a Line

Using Euler’s formula, find V if E = 30, F = 12.

Concept: Euler's Formula

The ratio of the areas of two triangles with common base is 6:5. Height of the larger triangle of 9 cm, then find the corresponding height of the smaller triangle.

Concept: Properties of Ratios of Areas of Two Triangles

In the following figure, point ‘A’ is the centre of the circle. Line MN is tangent at point M. If AN = 10 cm and MN = 5 cm, determine radius of the circle.

Concept: Theorem - Converse of Tangent at Any Point to the Circle is Perpendicular to the Radius

Draw `angle ABC` of measure 80° and bisect it

Concept: Basic Geometric Constructions

if `cos theta = 5/13` where `theta` is an acute angle. Find the value of `sin theta`

Concept: Trigonometric Identities

The volume of a cube is 512 cm^{3}. Find its side.

Concept: Surface Area of a Combination of Solids

The radius and slant height of a cone are 5 cm and 20 cm respectively. Find its curved surface area. (Π = 3.14)

Concept: Surface Area of a Combination of Solids

In the following figure, seg DH ⊥ seg EF and seg GK ⊥ seg EF. If DH = 18 cm, GK = 30 cm and `A(triangle DEF) = 450 cm^2`, then find:

1) EF

2) `A(triangle GFE)`

3) `A(square DFGE)`

Concept: Similarity of Triangles

In the following figure, ray PA is a tangent to the circle at A and PBC is a secant. If AP = 15, BP = 10, then find BC.

Concept: Tangent - Secant Theorem

Draw the circle with centre C and radius 3.5 cm. Point B is at a distance 7 cm from the centre. Draw tangents to the circle from the point B.

Concept: Construction of Tangent to the Circle from the Point on the Circle

Show that `sqrt((1+cosA)/(1-cosA)) = cosec A + cot A`

Concept: Trigonometric Identities

Write the equation of the line passing through A(–3, 4) and B(4, 5) in the form of ax + by + c = 0

Concept: Standard Forms of Equation of a Line

Prove that “The lengths of the two tangent segments to a circle drawn from an external point are equal.”

Concept: Number of Tangents from a Point on a Circle

A tree is broken by the wind. The top of that tree struck the ground at an angle of 30° and at a distance of 30. Find the height of the whole tree

Concept: Heights and Distances

A (5, 4), B (-3, -2) and C (1, -8) are the vertices of a triangle ABC. Find the equations of the median AD and line parallel to AC passing through the point B.

Concept: General Equation of a Line

Prove that, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of remaining two sides

Concept: Pythagoras Theorem

ΔSHR ~ ΔSVU. In ΔSHR, SH = 4.5 cm, HR = 5.2 cm, SR = 5.8 cm and `(SH)/(SV)=3/5`. Construct ΔSVU.

Concept: Division of a Line Segment

If ‘V’ is the volume of a cuboid of dimensions a × b × c and ‘S’ is its surface area, then prove that: `1/V = 2/S[1/a +1/b + 1/c]`

Concept: Surface Area of a Combination of Solids