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Question 1

Two line segments AB and AC include an angle of ${60}^{\circ }$, where AB = 5 cm and AC = 7 cm. Locate points P and Q on AB and AC, respectively such that $AP=\frac{3}{4}AB$ and $AQ=\frac{1}{4}AC$ .Join P and Q measure the length PQ.

Thinking Process

(i) Firstly we find the ratio of AB in which P divides, it with the help of the relation $AP=\frac{3}{4}AB$

(ii) Secondly, we find the ratio of AC in which Q divides. It with the help of the relation $AQ=\frac{1}{4}AC$.

(iii) Now, construct the line segment AB and AC in which P and Q respectively divide it in the ratio from step (i) and (ii) respectively.

(iv) Finally get the point P and Q. After that join PQ and get the required measurement of PQ.

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1 + $\frac{tanA}{tanB}$ =

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1) They can be congruent to each other.

2) They cannot be congruent to each other.

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Prove that two distinct lines cannot have more than one point in common.

In a right triangle ABC right-angled at C, P and Q are the points on the sides CA and CB respectively, which divide these sides in the ratio 2:1.





Prove that [4 MARKS]



(i) $9A{Q}^2=9{AC}^2+4{BC}^2$



(ii) $9{BP}^2=9{BC}^2+4{AC}^2$



(iii) $9({AQ}^2+{BP}^2)=13{AB}^2$

Can you think of any scientists, who have also been statesmen?

Question 127

Fill in the blank to make the statement true.

2916 x .............. = 0

The figure below shows a circle divided into 3 equal parts. The shaded region corresponds to ${\frac{2}{3}}^{rd}$ of the circle. If the unshaded region corresponds to ${\frac{a}{3}}^{rd}$ of the circle. Then, the value of a is ___.

If dividend = $x^4+x^3-2x^2+x+1$, divisor = x - 1 and remainder = 2. Find the quotient q(x).

Iron occurs as BCC as well as FCCunit cell. If the effective radius of an atom of iron is $124pm$, compute the density of iron in both these structures.

How many degrees are there in an angle which equals one-fifth of its supplement?

In the given figure, quadrilateral ABCD is circumscribed. The circle touches the sides AB, BC, CD and DA at P, Q, R, S respectively. If AP = 9 cm, BP = 7 cm, CQ = 5 cm and DR = 6 cm, find the perimeter of quad. ABCD.

A hollow prism with a square base of side 16 cm contains water 10 cm high. If a solid cube of side 8 cm is immersed in it, what will be the new height of water level in the prism?