Perpendicular from Right Angle to Hypotenuse Divides the Triangle into Two Similar Triangles

Q.

In triangle $\mathrm{ABC}$, $\mathrm{AD}$ is perpendicular to $\mathrm{BC}$ and ${\mathrm{AD}}^2=\mathrm{BD}.\mathrm{CD}$.

Prove that $△\mathrm{ABC}$ is a right triangle.

Q. proof that the external bisectors of any two angles of a triangle are concurrent with the internal bisector of the third angle

Q. in triangle pqr if pq =qr and L, M and, N are the mid points of the sides PQ, QR and RP respectively thanprove that LN=MN

Q. D is a point on side BC of ΔABC, such that AD = AC
Show that AB > AD.

Q.

The acute angles of a right-angled triangle are in the ratio of $2:3$. Find the measures of these angles.

Q. If two right-angled triangles ABC and PRQ are such that ∠A = 20°, ∠Q = 20°, and AC = QP, then which of the following is true?

1. ΔABC ≌ ΔPRQ

2. ΔABC ≌ ΔRQP

3. ΔABC ≌ ΔPQR

4. ΔABC ≌ ΔQRP

Q. In the given figure, PQR is a straight line. Find 𝑥, then complete the following:​(iii) $\angle AQR=$

Q. $△ABC$ is an isosceles triangle in which $AB=AC$. Sides $BA$ is produced to $D$ such that $AD=AB$. Show that $\angle BCD$ is a right angle.

Q. In the given figure, PQR is a straight line. Find 𝑥, then complete the following:​
(i) $\angle AQB=$

Q.

In △ ABC right-angled at B, a perpendicular is drawn from B which meets AC at M. If the ratio of areas of △ ABC and △ AMB is 9:4, find $\frac{AC}{AB}$ .

1. $\frac{9}{4}$
2. $\frac{3}{2}$
3. $\frac{2}{3}$
4. $\frac{4}{9}$

Q. In triangle $ABC,AD$ is perpendicular to $BC$ and $A{D}^2=BD\times DC$. Prove that angle $BAC={90}^0$.

Q. $ABC$ is an isosceles triangle with $AB=AC$. Draw $AP\perp BC$. Show that $\angle B=\angle C$

Q.

The angles of a triangle are in the ratio of $1:3:5$. Find the measure of each one of the angles.

Q. In the given diagram , $ABC$ is a staright line
(ii) If $y=1\frac{1}{2}$ right angle ; find $x$.

Q. Through a point on the hypotenuse of a right triangle lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is m times the area of the square. The ratio of the area of the other small right triangle to the area of the square is:

1. $m$
2. $\frac{1}{4m}$
3. $\frac{1}{2m+1}$
4. $\frac{1}{8m^2}$

Q. In an acute-angled triangle ABC, the internal bisector of angle A meets base BC at point D. DE $\perp$ AB and DF $\perp$ AC; hence AEF is an isosceles triangle.
State whether the above statement is true or false.

1. True
2. False

Q.

In the figure below, ray OS stands on a line POQ. Ray OR and ray OT are angle bisectors of ∠POS and ∠SOQ, respectively. If ∠POS = x, find ∠ROT.

1. ${120}^{\circ }$

2. ${180}^{\circ }$

3. ${90}^{\circ }$

4. ${75}^{\circ }$

Q. Area of any scalene triangle is given by the formula:

1. $\frac{1}{2}\times base\times height$
2. $\frac{\sqrt{3}}{4}\times (side{)}^2$
3. None of the above
4. $\frac{1}{4}\times base\times \sqrt{4(side{)}^2-(base{)}^2}$

Q. Write the converse of the Pythagoras Theorem and prove it.

Q. In a triangle, exterior angle measures 120°. If one of the interior opposite angle measure 50°, then what is the value of the other interior opposite angle?

Q. If in $\Delta ABC$ and $\Delta DEF,\frac{AB}{DE}=\frac{BC}{FD}$, then they will be similar if :

1. $\angle B=\angle D$
2. $\angle B=\angle E$
3. $\angle A=\angle D$
4. $\angle A=\angle F$

Q. $ABCD$ is a quadrilateral such that diagonal $AC$ bisects the angles $\angle A$ and $\angle C$ prove that $AB=AD$ and $CB=CD$.

Q. In the following figure, AB = AC and AD is perpendicular to BC. BE bisects angle B and EF is perpendicular to AB. Prove that FB = DC.

Q. In given figure $ABC$ and $DBC$ are two isosceles triangles on the same base $BC$. Show that $\angle ABD=\angle ACD$.

Q. In figure , $\angle MNP={90}^o,segNQ\perp segMP,MQ=9$cm , $QP=4$cm, find $NQ$

Q. $△$ ABC is right angled at B and the perpendicular drawn to the opposite side bisects it at D. BD = ___ cm if AD = DC = 5 cm.

Q.

The value of x in the given figure is ______.

1. 30^o
2. 50^o
3. 80^o
4. 40^o

Q. In the given figure, AB$=$AC, DB$=$DC then$\angle$ABD$=\angle$ACD.

1. True
2. False

Q. In a right angled triangle, the circumradius is half the:

1. Area
2. Perimeter
3. None of these
4. Hypotenuse

Q. In $\Delta ABC$, $D$ is a point on $BC$ such that $3BD=BC$. If each side of the triangle is $12cm$, then $AD$ equals:

1. $4\sqrt{5}cm$
2. $4\sqrt{6}cm$
3. $4\sqrt{7}cm$
4. $4\sqrt{11}cm$