Any Point Equidistant from the End Points of a Segment Lies on the Perpendicular Bisector of the Segment

Q.

In the given figure, AB = BC and AC = CD. Find the ratio of ∠BAD: ∠ADB.

1. 1: 3

2. 3 : 2

3. 3 :1

4. 2:3

Q.

If $a,b$ and $c$ are the sides of a triangle such that $a^4+b^4+c^4=2c^2(a^2+b^2)$, then the angles opposite to the side $c$ is

1. $45°$or $90°$

2. $30°$or $135°$

3. $45°$or $135°$

4. $60°$or $120°$

Q.

$AB$ is a line segment and $P$ is its mid-point. $D$ and $E$ are points on the same side of
$AB$ such that $\angle BAD=\angle ABE\text{and}\angle EPA=\angle DPB$ (See the given figure). Show that
$\Delta DAP\cong \Delta EBP$

Q.

Question 3
AD and BC are equal perpendiculars to a line segment AB.
Show that CD bisects AB.

Q. Question 10
In $\Delta ABC$ and $\Delta PQR$, if $AB=AC,\angle C=\angle P$ and $\angle B=\angle Q$. The two triangles are
(A) isosceles but not congruent
(B) isosceles and congruent
(C) congruent but not isosceles
(D) neither congruent nor isosceles

Q. Question 11
In $\Delta PQR$, $PD\perp QR$ such that D lies on QR. If PQ = a, PR = b, QD = c and DR = d, then prove that (a+b) (a-b) = (c+d) (c-d).

Q.

The length and breadth of a rectangle are $10cm$and $6cm$ respectively. Its area will be ,

1. $36c{m}^2$

2. $60c{m}^2$

3. $100c{m}^2$

4. $16c{m}^2$

Q. Question 2 (i)
AD is an altitude of an isosceles triangles ABC in which AB = AC. Show that
AD bisets BC

Q. Question 1 (ii)
$\Delta$ABC and $\Delta$DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see the given figure). If AD is extended to intersect BC at P, show that
$\Delta ABP\cong \Delta ACP$

Q.

In a $\Delta PQR,P{R}^2-P{Q}^2=Q{R}^2$and$M$is a point on side$PR$such that $QM\perp PR$. Prove that $Q{M}^2=PM\times MR$.

Q.

Find the area of the rectangles whose sides are: $2km$ and $3km$

Q.

An equilateral triangle ABC is inscribed in a circle with centre O. Then, $\angle BOC$ is equal to ___.

1. 30${}^{\circ }$
2. 120${}^{\circ }$
3. 60${}^{\circ }$
4. 90${}^{\circ }$

Q.

Find the area of the rectangles whose sides are: $2m$ and $70cm$

Q.

The locus of centers of all circles passing through two given points A and B is ______.

1. the perpendicular bisector of the line segment AB.

2. the bisector of the line segment AB.

3. any line perpendicular to the line segment AB.

4. a line parallel to the line segment AB.

Q.

A perpendicular bisector divides a line into two equal halves.

1. True

2. False

Q.

$\angle$OBC is ${90}^{\circ }$. Then, B must be the midpoint of CE. (True/False)

1. True

2. False

Q. If O is any point within $△$ABC, then to prove that AB + BC + CA > AO + BO + CO

Q. AD is an altitude of an isosceles triangles ABC in which AB = AC. Show that AD bisets BC.

Q.

In the given figure $ △ABC$ is a right-angled triangle at $ B$.

Let $ D$ and $ E$ be any points on $ AB$ and $ BC$ respectively.

Prove that $ {\left(AE\right)}^{2} + {\left(CD\right)}^{2} = {\left(AC\right)}^{2} + {\left(DE\right)}^{2}$

Q. The sides of a quadrilateral field, taken in order are $26m,27m,7m$ and $24m$ respectively. The angle contained by the last two sides is a right angle. Find its area.

Q.

In a $\Delta$ABC, it is given that AB = AC and AD is drawn perpendicular to BC. D is not the mid-point of BC.

1. True

2. False

Q. In $△ABC,m\angle B={90}^o,AB=4\sqrt{5},BD\perp AC,AD=4$, then $A\left(△ABC\right)=$?

1. $120$ sq.units
2. $160$ sq.units
3. $96$ sq.units
4. $80$ sq.units

Q. In the figure, ABC is triangle right angled at A and $AC\perp BD$
Show that
(i) $A{B}^2=BC.BD$
(ii) $A{C}^2=BC.DC$
(III) $A{D}^2=BD.CD$

Q. In Fig., $O$ is a point in the interior of a triangle $ABC,OD\perp BC,OE\perp AC$ and $OF\perp AB$. Show that
(i)$O{A}^2+O{B}^2+O{C}^2-O{D}^2-O{E}^2-O{F}^2=A{F}^2+B{D}^2+C{E}^2$,
(ii) $A{F}^2+B{D}^2+C{E}^2=A{E}^2+C{D}^2+B{F}^2$.

Q. $AD$ and $BC$ are equal perpendiculars to a line segment $AB$ (see figure). Show that $CD$ bisects $AB$.

Q.

In a right-angled triangle, one of the acute angles measures 53°. Find the measure of each angle of the triangle.

Q. The mid points of an equilateral triangle of side 10 cm are joined to form a triangle. What type of triangle is obtained by joining the mid points?
(3 marks)

Q. In given figure, E is a point on side CB produced of an isosceles triangle ABC with AB=AC. If $AD\perp BCandEF\perp AC$, prove that

Q. 18. angle CAB = 30 , angle ACB = 90 . ACFG and BCDE are the squares of side AC and BC respectively. Segment BG and Segment AE intersect AC and BG at K

Q. The locus of points, equidistant from two intersecting lines form the _________ between the lines.

1. angle bisector
2. perpendicular bisector
3. area
4. angle