Some Important Results

**Category : **9th Class

(i) The longer side of a triangle has greater angle opposite to it.

(ii) The greater angle of a triangle has longer side opposite to it.

(iii) Perpendicular line segment is the shortest in length. PM is the shortest line segment from point P to line I.

(iv) The distance between line and points lying on it is always zero.

(v) The sum of any two sides of a triangle is greater than the third side.

(vi) The difference between any two sides of a triangle is always less than its third side.

- Ellipses are not all similar to each other.
- Hyperbolas are not all similar to each other.
- In any triarigle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the Morley triangle.
- A non-planar triangle is a triangle which is not contained in a plane.

- Two triangle are said to be similar if their corresponding angles are equal and their corresponding sides are in proportion.
- If a line is drawn parallel to any one side of a triangle intersecting the other two sides at distinct point, then it divides the two sides in the same ratio, it is known as Thale's Theorem.
- The ratio of area of two similar triangles is equal to the ratio of the square of the corresponding sides.
- In a right angled triangle, the square of hypotenuse is equal to the sum of the square of the other two sides, it is known as Pythagoras Theorem.
- Two triangles are said to be congruent if and only if their corresponding sides and angles are equal.

**The quadrilateral which is formed by joining the mid points of a given quadrilateral is always_______.**

(a) A parallelogram

(b) A trapezium

(c) A rhombus

(d) A rectangle

(e) None of these

**Answer:** (a)

**Explanation:**

**Given:**

A quadrilateral LMNO is a quadrilateral in which P, Q, R and S are the mid points of LM, MN, NO, OL respectively.

**To Prove:**

PQRS is a parallelogram

**Construction:**

Join LN

**Proof:**

In\[\Delta \text{LMN}\],

\[PQ\,|\,\,|\,LN\] and \[\text{PQ}=\frac{1}{2}\text{LN}\] ...,.(i)

[Because P and Q are the mid points of LM and MN respectively]

In \[\Delta \text{LNO}\]

\[SR\,|\,\,|LN\] and \[\text{SR}=\frac{1}{2}\text{LN}\] .....(ii)

[Because S and R are the mid - points of LO and ON respectively]

From (i) and (ii), we get

\[\text{PQ}=\text{SR}\] and \[~\text{PQ}\,\text{ }\!\!|\!\!\text{ }\,\,\text{ }\!\!|\!\!\text{ }\,\text{SR}\] ....(iii)

Similarly we can see that

\[\text{PS}=\text{QR}\] and \[\text{PS}\,\text{ }\!\!|\!\!\text{ }\,\,\text{ }\!\!|\!\!\text{ }\,\text{QR}\] .....(iv)

From (iii) and (iv), we conclude that quadrilateral PQRS is a parallelogram.

**In \[\Delta \text{LMN}\] which LP is the bisector of MN and BR is the bisector of LP. Here, points P and R are the points of line segment MN and LN respectively then is equal to.... .**

(a) \[\frac{2}{3}LN\]

(b) \[\frac{1}{3}LN\]

(c) \[\frac{2}{3}LM\]

(d) \[\frac{1}{3}LM\]

(e) None of these

**Answer:** (b)

**Explanation:**

Let the Q is the midpoint of LP

Draw a line \[PS\,|\,\,|\,MR\] since P is the midpoint of MN

\[\Rightarrow \]S is the midpoint of RN

In\[\Delta \text{LPN}\],

\[QR\,|\,\,|\,\,PS\]

\[\Rightarrow \] R is the midpoint of \[LS\] \[\Rightarrow \]\[LR=RS=SN\] \[\Rightarrow \]\[LR=\frac{1}{3}(LN)\]

** In the figure given below \[\text{ON}\,\text{ }\!\!|\!\!\text{ }\,\,\text{ }\!\!|\!\!\text{ }\,\text{LM}\]**

**the value of\[x\]is.....**

(a) 5, 11

(b) 5, 8

(c) 8, 5

(d) 18, 5

(e) None of these

**Answer:** (d)

**In the given \[\Delta \text{DEF},\text{ GH }\!\!|\!\!\text{ }\,\,\text{ }\!\!|\!\!\text{ EF}\] and if \[\frac{DG}{GE}=\frac{3}{5}\] and DF = 5.6 cm then the value of DE is equal to..... .**

(a) 2.1cm

(b) 3.36cm

(c) 9.35cm

(d) 2.24cm

(e) None of these

**Answer:** (a)

**In a trapezium PQRS, PQ||RS. If the diagonal PR and QS intersect at a point 0 such that OP = 6 cm and OR = 8 cm, then the ratio of ar(\[\Delta \text{POQ}\]) and ar (\[\Delta ROS\]) is equal to:**

(a) \[\frac{3}{4}\]

(b) \[\frac{36}{48}\]

(c) \[\frac{9}{16}\]

(d) \[\frac{5}{16}\]

(e) None of these

**Answer:** (c)

** If the midpoint of the hypotenuse is joined with the vertex of a right angled triangle where the right angle formed then the length of this line segment is:**

(a) \[\frac{1}{2}\] of the base

(b) \[\frac{1}{2}\] of the perpendicular

(c) \[\frac{1}{2}\] of the hypotenuse

(d) \[\frac{1}{2}\] of the area of triangle

(e) None of these

**Answer:** (c)

**Explanation: **

**Given:**

The figure given below represents all the conditions S is the midpoint of PR,

**To prove:**

**Construction:**

Produce QS to T; so that QS = ST and Join RT.

**Proof:**

In\[\Delta \text{PSQ}\] and \[\Delta \text{RST}\] \[\text{PS}=\text{SR}\] (S is the mid point of PR)

\[\text{QS}=\text{ST}\] (By construction)

\[\angle \text{PSQ}=\angle \text{TSR}\] (Vertically opposite angles)

\[\therefore \]\[\Delta \text{PSQ}=\Delta \text{RST}\] (By S-A-S criteria)

\[\therefore \]\[\text{PQ}=\text{RT}\]and\[\angle \text{QPR}=\angle \text{PRT}\] (Alternate interior angles)

\[\Rightarrow \]\[\text{RT}\,\text{ }\!\!|\!\!\text{ }\,\,\text{ }\!\!|\!\!\text{ }\,\text{PQ}\] \[\Rightarrow \]\[\angle \text{TRQ}=\text{9}0{}^\circ \]

Now in\[\Delta \text{PQR}\]and\[\Delta \text{TRQ}\]

\[\text{QR}=\text{QR}\] (Common side)

\[\text{PQ}=\text{RT}\] (Already proved)

\[\therefore \] \[\Delta PQR\cong \Delta TRS\] (By R-H-S criteria)

\[\Rightarrow \]\[PR=QT\]\[\Rightarrow \]\[\frac{1}{2}QT=\frac{1}{2}PR\]

**In the figure given below:**

**LP, MQ and NR are the median of \[\Delta \text{LMN}\] then which one of the following conditions is correct?**

(a) \[(\text{LM}+\text{MN}+\text{NP})<(\text{LP}+\text{MQ}+\text{NR})\]

(b) \[\text{(LM}+\text{MN}+\text{NP})>(\text{NP}+\text{MQ}+\text{NR})\]

(c) \[\text{(LM}+\text{MN}+\text{NP})>(\text{LP}+\text{MQ}+\text{NR)}\]

(d) \[\text{(L}{{\text{M}}^{2}}+\text{M}{{\text{N}}^{2}}+\text{N}{{\text{P}}^{2}})>(\text{L}{{\text{P}}^{2}}+\text{M}{{\text{Q}}^{2}}+\text{N}{{\text{R}}^{2}}\text{)}\]

(e) None of these

**Answer:** (c)

**One pair of vertically opposite angles of a quadrilateral are a and b respectively as shown in the figure.**

**Then which one of the following relations is correct?**

(a) \[\text{a}+\text{b}=\text{c}+\text{d}\]

(b) \[\text{a}-\text{b}=\text{c}+\text{d}\]

(c) \[\text{a}+\text{b}=\text{c}-\text{d}\]

(d) \[\text{a}+\text{b}+\text{c}+\text{d}=\text{36}0{}^\circ \]

(e) None of these

**Answer:** (a)

**In the figure given below:**

**\[\angle \text{DEX}=\angle \text{YEX},\text{ DE}=\text{EX}\] and \[\text{EX}=\text{EF}\] then XY is equal to.....**

(a) \[\angle \text{DEX}\]

(b) DF

(c) \[\angle \text{XEF}\]

(d) EF

(e) None of these

**Answer:** (b)

** The line segment PQ and RS intersect each other at point 0 in such a way that OP = OS and OQ = OR then which one of the following options is correct?**

(a) PR = SQ and \[~\text{PR }\!\!|\!\!\text{ }\,\,\text{ }\!\!|\!\!\text{ SQ}\]

(b) OP=OQ and \[~\text{PR}\bcancel{\text{ }\!\!|\!\!\text{ }\,\,\text{ }\!\!|\!\!\text{ }}\text{SQ}\]

(c) RO = OS and\[~\text{PR }\!\!|\!\!\text{ }\,\,\text{ }\!\!|\!\!\text{ SQ}\]

(d) PR = SQ and PR and SQ may or may not be parallel to each other

(e) None of these

**Answer:** (d)

*play_arrow*Similarly*play_arrow*Similar Triangles*play_arrow*Congruency Figures*play_arrow*Some Important Results*play_arrow*Triangle

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