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Home Question Bank Math Geometry Archive 2024 December 31

Geometry Questions from Dec 31,2024

Browse the Geometry Q&A Archive for Dec 31,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Arc Length Find the arc length for the following. 13. A circle has a radius of 5.75 inches. Find the length of the arc intercepted by a central angle of \( \frac{5 \pi}{6} \). 14. A circle has a diameter of 10 inches. Find the length of the arc intercepted by a central angle of \( 125^{\circ} \). Figure 4.27 The areas of two similar triangles are \( 64 \mathrm{~cm}^{2} \) and \( 144 \mathrm{~cm}^{2} \), a) Find the ratio of their altitudes. b) If the perimeter of the smaller triangle is 49 cm , find the perimeter of the larger triangle. If two vertically opposite angles are supplementary, then the measure of each angle of them equals \( \begin{array}{llll}\text { (a) } 45^{\circ} & \text { (b) } 60^{\circ} & \text { (c) } 90^{\circ} & \sqrt{3}\end{array} \) 1 Choose the correct answer from those given: \( \begin{array}{llll}\text { (1) The distance between the point }(-4,-3) \text { and the } X \text {-axis equals ............ length u } \\ \begin{array}{llll}\text { (a) }-3 & \text { (b) } 3 & \text { (c) } 4 & \text { (d) }-4\end{array}\end{array} l^{\circ} \) 1. If a rectangle has a length of 10 m and a height of 5 m , what is its perimeter? 8. Find the volume of the box (parallelepiped) determined by the points \( P=(1,2,4) \), \( Q=(2,-5,8), R=(4,2,-3) \) and \( S=(3,5,-6) \). 9. Find the volume of the box (parallelepiped) determined by the points \( P=(8,5,3) \), \( Q=(4,6,7), R=(-4,-2,9) \) and \( S=(3,-3,-4) \). 6. Find the volume of the box (parallelepiped) determined by \( u=4 i+5 j-2 k \), \( v=2 i+3 j+5 k \) and \( w=5 i+4 j+8 k \) 7. Find the volume of the box (parallelepiped) determined by \( u=2 i+6 j-3 k \), \( v=5 i-4 j-7 k \) and \( w=-7 i-4 j+2 k \) 3. Find a vector perpendicular to the plane of using the points \( P=(1,2,-4) \), \( Q=(2,5,-7) \) and \( R=(3,-5,-1) \). Also find area of the parallelogram and area of triangle. Furthermore find a unit vector perpendicular to the plane using these points. 1. If \( u=3 i+6 j-7 k, v=4 i-5 j+6 k \) and \( w=-2 i+3 j-9 k \) then use the algebra of vectors find the followings \[ \begin{array}{l} (3 u-4 v) \bullet(5 w-2 u) \\ (2 v+3 w) \bullet(4 u-3 v) \\ (u+v+3 w) \bullet(u-v-2 w) \\ (2 u+v) \times(v-2 w) \\ (w+u) \times(w-u) \\ (v+u) \times(2 w-3 u) \end{array} \] 2. Find a vector perpendicular to the plane of using the points \( P=(2,4,6) \), \( Q=(1,2,9) \) and \( R=(1,-1,7) \). Also find area of the parallelogram and area of triangle. Furthermore find a unit vector perpendicular to the plane using these points. 3. Find a vector perpendicular to the plane of using the points \( P=(1,2,-4) \), \( Q=(2,5,-7) \) and \( R=(3,-5,-1) \). Also find area of the parallelogram and area of triangle. Furthermore find a unit vector perpendicular to the plane using these points. 4. If \( u=3 i+6 j-7 k, v=4 i-5 j+6 k \) and \( w=-2 i+3 j-9 k \) then show that \( (u \times v) \bullet w=(v \times w) \bullet u=(w \times u) \bullet v \) 5. If \( u=2 i+4 j+8 k, v=2 i-3 j-6 k \) and \( w=5 i+4 j+8 k \) then show that \( (u \times v) \bullet w=(v \times w) \bullet u=(w \times u) \bullet v \) 6. Find the volume of the box (parallelepiped) determined by \( u=4 i+5 j-2 k \), \( v=2 i+3 j+5 k \) and \( w=5 i+4 j+8 k \) 7. Find the volume of the box (parallelepiped) determined by \( u=2 i+6 j-3 k \), \( v=5 i-4 j-7 k \) and \( w=-7 i-4 j+2 k \) REVIEW EXERCISE FOR UNIT 4 Wrte true if the statement is correct and false if it is ont. a) Two congruent polygons are necessarily similar. b) Two similar polygons are necessarily congruent. c) Congruence is a similarity when the constant of proporicolity ts d) Rectangle ABCD is similar to rectangle EFGH if \( \frac{A B}{E F}=\frac{B C}{F G} \) c) Doubling the side length of a triangle doubles the area. f) Two tnangles with the same area are similar
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