A student makes the following statement It looks like if you

A student makes the following statement: \"It looks like if you join the midpoints of the sides of 116. A student asks, \"Is the area of a geometric (two-dimensional) figure always smaller than its 117. A student asks you if it is possible for a figure to be symmetric about a point but not symmetric 118. A student asks, \"If a square ABCD is folded on either of its diagonals, it is mapped onto itself. 15. a quadrilateral, the resulting inscribed figure is always a parallelogram.\" perimeter? Likewise, is the volume of a solid always smaller than its surface area?\" about any line. Does this occur for a rhombus which is not a square?\" 119. A student asks you if there is such a thing as equivalent triangles. 20. A student asks, \"Is is possible for one triangle to have five parts (angles, sides) congruent to five parts of another triangle and still not have the two triangles congruent?\"

Solution

Ans 1)

let ABCD is quadrilateral amd having midpoints M, N,P,Q

To prove that MNPQ is a paralleogram

that is we have to show that their opposite sides are parallel

that os we have to show that MN is parallel to PQ and QM is parallel to NP

Proof :

first we join B and D

means draw BD

To joining this two points we get two triangle ABD and triangle BCD

Here MN is the midsegment of triangle ABD

so MN is parallel to BD

QP is also midsegment of triangle BCD

so QP is parallel to BD

now MN is parallel to BD and BD is parallel to QP

By transitivity ,

MN is parallel to QP

Secondaly join A and C

means we have to draw AC

We get also two triangle ADC and ABC

MQ is midsegment of triangle ACD

so MQ is parallel to AC

NP is midsegment of triangle ABC

so NP is parallel to AC

since MQ is parallel to AC and AC is parallel to NP

By transitivity,

MQ is parallel to NP

since MN is parallel to QP and MQ is parallel to NP

Therefore by defination of parallelogram

MNPQ is a parallelogram

so we say that the \"if join the midpoint of quadrilateral then it forms parallelogram\"

Hence proved

Done