Cho hình thang ABCD có diện tích là 360cm2. Thêm điểm M ở cạnh AB, MB =\(\frac{1}{3}\)AM. Thêm điểm N ở giữa cạnh BC, BN =\(\frac{1}{3}\) NC, thêm điểm P ở giữa cạnh CD, PD = \(\frac{1}{3}\)CP, thêm điểm Q ở giữa cạnh DA, DQ=\(\frac{1}{3}\) QA. Diện tích MNPQ bằng bao nhiêu xăng - ti - mét vuông ?
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\(S_{BMN}=\dfrac{1}{2}xBMxBN=\dfrac{1}{2}x\dfrac{AB}{4}x\dfrac{BC}{2}=\dfrac{1}{16}xS_{ABCD}\)
\(S_{CPN}=\dfrac{1}{2}xCNxCP=\dfrac{1}{2}x\dfrac{BC}{2}x\dfrac{CD}{2}=\dfrac{1}{8}xS_{ABCD}\)
\(S_{DPQ}=\dfrac{1}{2}xPDxDQ=\dfrac{1}{2}x\dfrac{CD}{2}x\dfrac{AD}{3}=\dfrac{1}{12}xS_{ABCD}\)
\(S_{AMQ}=\dfrac{1}{2}xAMxAQ=\dfrac{1}{2}x\dfrac{3xAB}{4}x\dfrac{2xAD}{3}=\dfrac{1}{4}xS_{ABCD}\)
\(\Rightarrow S_{MNPQ}=S_{ABCD}-\left(S_{BMN}+S_{CPN}+S_{DPQ}+S_{AMQ}\right)\)
Bạn tự thay số rồi tính nốt nhé
SAMQ = \(\dfrac{1}{2}\)AM\(\times\)AQ = \(\dfrac{1}{2}\times\) \(\dfrac{1}{2}\)AB\(\times\)\(\dfrac{1}{2}\)AD = \(\dfrac{1}{8}\)\(\times\)SABCD
SDPQ = \(\dfrac{1}{2}\)DQ\(\times\)DP = \(\dfrac{1}{2}\)\(\times\)\(\dfrac{1}{2}\) AD\(\times\)\(\dfrac{1}{2}\)DP = \(\dfrac{1}{8}\) \(\times\) SABCD
CN = CB - BN = CB - \(\dfrac{1}{3}\)CB = \(\dfrac{2}{3}\)CB
SCPN = \(\dfrac{1}{2}\)CP\(\times\)CN = \(\dfrac{1}{2}\)\(\times\)\(\dfrac{1}{2}\) CD\(\times\)\(\dfrac{2}{3}\)CB = \(\dfrac{1}{6}\)SABCD
SBNM = \(\dfrac{1}{2}\)BN\(\times\)BM = \(\dfrac{1}{2}\)\(\times\)\(\dfrac{1}{2}\)AB\(\times\)\(\dfrac{1}{3}\)BC = \(\dfrac{1}{12}\)SABCD
Diện tích tứ giác MNPQ bằng: (1 - \(\dfrac{1}{8}\) - \(\dfrac{1}{8}\) - \(\dfrac{1}{6}\) - \(\dfrac{1}{12}\) )SABCD = \(\dfrac{1}{2}\)SABCD
Diện tích của tứ giác MNPQ là: 240\(\times\)\(\dfrac{1}{2}\) = 120 (cm2)
AM = BM = \(\dfrac{1}{2}\)AB; AQ = QD = \(\dfrac{1}{2}\) AD
SAMQ = \(\dfrac{1}{2}\)AM\(\times\)AQ =\(\dfrac{1}{2}\times\) \(\dfrac{1}{2}\)AB\(\times\)\(\dfrac{1}{2}\)AD = \(\dfrac{1}{8}\)SABCD = 240\(\times\)\(\dfrac{1}{8}\)=30(cm2)
DQ = QA = \(\dfrac{1}{2}\)AD; DP = PC = \(\dfrac{1}{2}\) DC
SDPQ =\(\dfrac{1}{2}\times\)DP\(\times\) DQ =\(\dfrac{1}{2}\) \(\times\)\(\dfrac{1}{2}\)AD\(\times\)\(\dfrac{1}{2}\)DC =\(\dfrac{1}{8}\)SABCD = 240\(\times\)\(\dfrac{1}{8}\)=30(cm2)
CN = BC - BN = BC - \(\dfrac{1}{3}\)BC = \(\dfrac{2}{3}\)BC
SCPN = \(\dfrac{1}{2}\)CP\(\times\)CN= \(\dfrac{1}{2}\)\(\times\) \(\dfrac{1}{2}\)CD \(\times\) \(\dfrac{2}{3}\) BC = \(\dfrac{1}{6}\)SABCD=240\(\times\dfrac{1}{6}\)=40 (cm2)
SBMN=\(\dfrac{1}{2}\) BM\(\times\)BN =\(\dfrac{1}{2}\times\dfrac{1}{2}\)AB\(\times\)\(\dfrac{1}{3}\)BC=\(\dfrac{1}{12}\)SABCD=240\(\times\)\(\dfrac{1}{12}\)=20(cm2)
Diện tích tứ giác MNPQ là:
240 - (30 + 30 + 40 + 20) = 120(cm2)
Đáp số: 120 cm2
minh chiu