B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई chapter 10 - Mid-point Theorem [Latest edition]

In ΔABC, M and N are mid-points of sides AB and AC respectively.

1. If MN = 4x – 2 and BC = 6x + 3, find x.
2. If ∠MNC = (3y + 10)° and ∠C = (y + 10)°, find y. 
3. If ∠AMN = (2a + 15)° and ∠B = (3a – 15)°, find a. 
4. If AB = 7 cm, BC = 10 cm, AC = 9.2 cm, find the perimeter of MNCB.

In ΔABC, ∠B = 90°. D is the mid-point of AB and DE || BC. If AB = 9 cm and AC = 15 cm, find the perimeter of DECB.

In ΔXYZ, M and N are mid-points of XY and XZ respectively. P and Q are mid-points of XM and XN. If PQ = 2.8 cm, find the length of YZ.

In trapezium ABCD, M and N are mid-points of AC and BD, and AB || CD. Find

1. MN if AB = 5.5 cm, CD = 7.5 cm
2. AB if CD = 16 cm, MN = 10 cm
3. CD if AB = 14 cm, MN = 16.5 cm

AB || CD || EF and AC = CE. Find x if

1. BD = 3x + 2 and DF = 4x – 3
2. BD = 5x – 4 and BF = 4x + 16

In ΔABC, M and N are mid-points of AB and AC, and NP || AB.

1. Prove that BMNP is a parallelogram.
2. If AB = 11 cm, BC = 12 cm, find the perimeter of BMNP.

In ΔPQR, ∠Q = 90°. QM is the median of the triangle through Q. Prove that QM = `1/2` PR.

Medians CM and BN of ΔABC are produced to P and Q respectively such that CM = MP and BN = NQ. Prove that the points P, A, Q are collinear and PA = AQ.

[Hint: Join MN.MN = `1/2` PA and || to PA in ΔCAP, MN = `1/2` AQ and || to AQ in ΔBAQ, etc]

Prove that the three line segments which join the mid-points of the sides of triangle, divide it into four triangles which are congruent to each other.

Prove that the triangle formed by joining the mid-points of the sides of an equilateral triangle is also equilateral.

In ΔABC, M, N and P are mid-points of sides AB, AC and BC respectively. X, Y and Z are mid-points of sides of ΔMNP. If XY = 2.5 cm, YZ = 3.5 cm and XZ = 4 cm, find the sides of ΔABC.

ABCD is a parallelogram. M is the mid-point of AB and P is a point on diagonal BD such that BP = `1/4` BD. MP produced meets BC at N. Prove that:

1. N is a mid-point of BC.
2. MN = `1/2` AC

In ΔABC, AM is a median and N is the mid-point of AM. BN produced meets AC at P. Prove that AP = `1/3` AC.

[Hint: Through M draw MQ parallel to BP.]

ABCD is a rectangle. P, Q, R and S are mid-points of sides of the rectangle as shown in the given figure. Prove that PQRS is a rhombus.

[Hint: Join DB. Use mid-point theorem in ΔADB and ΔCDB to show PS = `1/2` DB and PS || to DB, etc.]

ABCD is a parallelogram, E and F are the midpoints of AB and CD respectively. GH is any line that intersects AD, EF and BC in G, P and H respectively. Prove that GP = PH.

[Hint: Use Intercept Theorem on AD || EF || BC.]

ABC is a right-angled triangle with hypotenuse AC = 13 cm and side AB = 5 cm. Perpendiculars are drawn from mid-point M of AC to AB and BC. What is the perimeter of the resulting quadrilateral?

In the quadrilateral ABCD, AD || BC. P and Q are mid-points of AB and AC. Prove that

1. R is the mid-point of DC.
2. AD + BC = 2PR

In parallelogram PQRS, M is the mid-point of PQ. PT drawn parallel to MR meets SR at N and QR produced at T. Prove that

1. PS = `1/2` QT
2. PT = 2MR

ABCD is a kite in which AB = AD and BC = DC. M, N and O are mid-points of sides AB, BC and CD. Prove that

1. ∠MNO = 90°
2. The line MP drawn parallel to NO bisects AD.

In ΔABC, AM = MC = BM. Prove that ∠ABC = 90°.

In ΔPQR, PQ = 5 cm, QR = 12 cm, PR = 13 cm. M and N are mid-points of PQ and QR. The length of MN is ______.

In ΔABC, M and N are midpoints of AB and AC. ∠AMN = 3x – 25, ∠B = 2x + 5.
∴ The value of x is:

M and N are mid points of sides AD and BC in the trapezium ABCD. AB = 6.5 cm, MN = 8 cm.

∴ DC = ........

PQ || MN || SR and PM = MS. Find x if QN = 3x + 1 and NR = 2x + 6.

M and N are mid points of AB and AC of ΔABC. AB = 9 cm, BC = 12 cm. Perimeter of BPNM is ______.

In ΔABC, M, N, P are midpoints of sides AB, AC and BC. X, Y, Z are midpoints of sides of ΔMNP. If XY = 2 cm, YZ = 2.5 cm, XZ = 3 cm, then sides of ΔABC are ______.

In ΔPQR, ∠P = 90°, PQ = 8 cm, QR = 17 cm. From midpoint M perpendiculars are drawn parallel to sides PQ and PR. Perimeter of rectangle PAMB is ______.

When the midpoints of sides of a kite are joined in order, the quadrilateral formed is a ______.

When the midpoints of a rectangle are joined in order, the quadrilateral formed is a ______.

When the midpoints of a quadrilateral are joined in order, the figure formed is a ______.

A square is formed when the midpoints of ______ are joined in order.

ABCD is a rhombus, O is the midpoint of BC. AD = 6 cm. DP is ______.

Name the 2 quadrilaterals K₁ and K₂ when midpoints of K₁ are joined in order K₂ is formed and when midpoints of K₂ are joined K₁ is formed.

In the rectangle ABCD, AB = 9 cm, BC = 12 cm. Midpoints M and N of AB and BC are joined. Length of MN is ______.

M and N are midpoints of sides AB and BC of ΔABC. If AC = 12 cm and ∠BMN = 55°. Then

1. MN = 6 cm
2. MN || AC
3. ∠c = 55°

Which statements from the above are valid?

Assertion: In ΔABC, AP ⊥ BC. E and F are midpoints of AB and AC, then AQ = QP.

Reason: Q is the midpoint of AP from midpoint theorem.

Assertion: ABCD and PQRC are rectangles. Q is the midpoint of AC, then BP = PC.

Reason: Through the midpoint of one side of a triangle, a line is drawn parallel to another side it bisects the third side.

Assertion: In ΔABC, M and N are midpoints of sides AB and AC. ∠BMN = 6x and ∠B = 2x. ∴ x = 22.5°.

Reason: Co-interior angles of parallel lines are supplementary.

In the figure, lines l, m and n are parallel. AP = PQ. Find

1. BC if AB = 3.5 cm
2. SQ if RQ = 2.8 cm
3. CQ if BP = 3.2 cm
4. PR if AS = 7.4 cm
5. BR if AS = 6.5 cm and CQ = 9.5 cm

M and N are mid-points of AB and AC.

1. Find x if ∠MNC = 3x – 10° and ∠C = x + 18°
2. Find y, if MN = (2y + 3) cm and BC = (3y + 8) cm

∠ABC = 90°. ABP is an equilateral triangle. PQ || BC. Prove that

1. PQ ⊥ AB
2. AQ = QB
3. R is the mid-point of AC.

ABCD is a parallelogram. P is the mid-point of DC and Q is a point on AC such that CQ = `1/4` AC. PQ produced meets BC at R. Prove that R is the mid-point of BC. [Hint: Join DB.]

PQRS is a parallelogram. M and N are mid-points of PQ and QR. Diagonals PR and QS meet at O.

1. Prove that MONQ is a parallelogram.
2. If PQ = 8 cm and PS = 6 cm, find the perimeter of MONQ.

In ΔABC, M and N are mid-points of sides AB and BC. P is a point on AC such that PN || AB. Prove that PMBN is a parallelogram.

ABCD is a rectangle. P, Q, R and S are the mid-points of sides AB, BC, CD and AD respectively. What kind of figure is PQRS? If AB = 32 cm and BC = 24 cm, find the perimeter of PQRS.

In ΔABC, M and N are mid-points of AB and AC respectively. P is any point on BC. Prove that MN bisects AP.