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7th Class Mathematics The Triangle and its Properties

The Triangle and its Properties

1) In \[\Delta PQR\], D is the mid-point of\[QR\]. PM is ?????? PD is ??????. Is \[QM=MR?\]
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2) Draw rough sketches for the following : (a) In \[\Delta ABC\], BE is a median. (b) In \[\Delta PQR,PQ\] and PR are altitudes of the triangle. (c) In \[\Delta XYZ,YL\] is an altitude in the exterior of the triangle.
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3) Verify by drawing a diagram if the median and altitude of an isosceles triangle can be same.
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4) Find the value of the unknown exterior angle \[x\] in the following diagrams: (i) (ii) (iii) (iv) (v) (vi)
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5) Find the value of the unknown interior angle \[x\] in the following figures: (i) (ii) (iii) (iv) (v) (vi)
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6) Find the value of the unknown \[x\] in the following diagrams: (i) (ii) (iii) (iv) (v) (vi)
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7) Find the values of the unknowns \[x\] and \[y\] in the following diagrams: (i) (ii) (iii) (iv) (v) (vi)
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8) Is it possible to have a triangle with the following sides? (i) \[\text{2 cm},\text{ 3 cm},\text{ 5 cm}\] (ii) \[\text{3 cm},\text{ 6 cm},\text{ 7 cm}\] (iii) \[\text{6 cm},\text{ 3 cm},\text{ 2 cm}.\]
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9) Take any point O in the interior of a triangle PQR. Is (i) \[OP+OQ>PQ?\] (ii) \[OQ+OR>QR?\] (iii) \[OR+OP>RP?\]
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10) AM is a median of a triangle ABC. Is \[AB+BC+CA>2AM?\] (Consider the sides of triangles \[\Delta ABM\] and \[\Delta AMC\].)
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11) \[ABCD\] is a quadrilateral. Is \[AB+BC+CD+DA<2(AC+BD)\]?
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12) The lengths of two sides of a triangle are 12 cm and 15 cm. Between what two measures should the length of the third side fall?
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13) PQR is a triangle right-angled at P. If \[PQ=\text{10 cm}\] and \[\text{PR}=\text{24 cm}\], find \[\text{QR}\].
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14) ABC is a triangle right-angled at C. If \[\text{AB}=\text{25 cm}\] and \[\text{AC}=\text{7 cm},\] find BC
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15) A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.
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16) Which of the following can be the sides of a right triangle? (i) \[\text{2}.\text{5 cm}\], \[\text{6}.\text{5 cm}\], \[\text{6 cm}\]. (ii) \[\text{2 cm}\], \[\text{2 cm}\], \[\text{5 cm}\]. (iii) \[\text{1}.\text{5 cm}\], \[\text{2 cm}\], \[\text{2}.\text{5 cm}\]. In the case of right-angled triangles, identify the right angles.
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17) A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base of the tree. Find the original height of the tree.
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18) Angles Q and R of a \[\Delta PQR\] are \[\text{2}{{\text{5}}^{o}}\] and \[\text{6}{{\text{5}}^{o}}\]. Write which of the following is true: (i) \[P{{Q}^{2}}+Q{{R}^{2}}=R{{P}^{2}}\] (ii) \[P{{Q}^{2}}+R{{P}^{2}}=Q{{R}^{2}}\] (iii) \[R{{P}^{2}}+Q{{R}^{2}}=P{{Q}^{2}}\]
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19) Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.
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20) The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.
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